#72341198

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233559	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(2,8)(3,14)(4,20)(5,24)(10,36)(12,13)(16,47)(18,19)(21,53)(22,23)(27,28)(31,39)(33,45)(34,40)(37,59)(38,57)(41,56)(43,49)(46,61)(48,55)(50,64)(52,54)(58,63)(60,62);(1,2)(3,15,18,30)(4,10,22,31)(5,25,24,42)(6,12)(7,13)(8,29)(9,14,35,19)(11,27,32,28)(16,37,49,59)(17,40,51,61)(20,39,23,36)(21,41)(26,46,44,34)(33,47,45,43)(38,53)(48,60,64,62)(50,63,55,58)(52,56)(54,57);(1,3)(2,9,8,30)(4,16)(5,26,27,44)(6,18)(7,19)(10,37,31,33)(11,40,25,46)(12,35,13,15)(14,29)(17,24,51,28)(20,49)(21,48,52,55)(22,43)(23,47)(32,34,42,61)(36,45,39,59)(38,63,41,58)(50,53,64,54)(56,60,57,62);(1,4)(2,10,8,31)(3,16)(5,21,27,52)(6,23)(7,20)(9,33,30,37)(11,41,25,38)(12,36,13,39)(14,43)(15,45,35,59)(17,50,51,64)(18,47)(19,49)(22,29)(24,54,28,53)(26,55,44,48)(32,57,42,56)(34,60,61,62)(40,63,46,58);(1,5,7,27)(2,11)(3,17,18,26)(4,21)(6,28,29,24)(8,32)(9,34,30,40)(10,38,39,57)(12,25)(13,42)(14,44,19,51)(15,46,35,61)(16,48,43,50)(20,52)(22,54)(23,53)(31,56,36,41)(33,58,59,63)(37,62,45,60)(47,64,49,55)

Polycyclic presentation:

2^10|b^2=f,c^2=g,d^2=g,e^2=ghj,b^a=bij,c^a=cij,c^b=ch,d^a=dj,d^b=dfg,d^c=dg,e^a=eij,e^b=eg,e^c=efgh,e^d=eh,f^c=fi,f^d=fij,f^e=fij,g^b=gij,g^e=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341200
maximal normal subgroup of	#72341200

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);