#72341049

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

2578 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)(9,15)(10,38)(11,34)(12,13)(17,27)(18,19)(21,33)(23,52)(24,25)(26,43)(32,37)(35,53)(36,45)(39,58)(40,61)(41,48)(42,54)(50,51)(56,57)(59,63)(60,62);(1,2)(3,15)(4,21,24,33)(5,26,28,43)(6,12)(7,13)(8,31)(9,18)(10,25,38,20)(11,29,34,30)(14,32)(16,35,47,39)(17,48,51,45)(19,37)(22,53,44,58)(23,54,52,40)(27,36,50,41)(42,57,61,56)(46,62,49,59)(55,60,64,63);(1,3,6,14)(2,9,12,37)(4,22,25,16)(5,27,30,51)(7,18,31,19)(8,32,13,15)(10,39,33,58)(11,41,26,48)(17,28,50,29)(20,47,24,44)(21,53,38,35)(23,55)(34,36,43,45)(40,59)(42,63)(46,57)(49,56)(52,64)(54,62)(60,61);(1,4)(2,10,8,33)(3,16,14,44)(5,23,30,56)(6,20)(7,24)(9,35,15,39)(11,42,43,61)(12,21,13,38)(17,49,51,64)(18,47,19,22)(25,31)(26,54,34,40)(27,46,50,55)(28,57,29,52)(32,53,37,58)(36,60)(41,62)(45,59)(48,63);(1,5,7,30)(2,11,8,34)(3,17)(4,23,25,57)(6,29,31,28)(9,36,37,48)(10,40,38,61)(12,43,13,26)(14,27)(15,45,32,41)(16,46,44,49)(18,50)(19,51)(20,52,24,56)(21,54,33,42)(22,55,47,64)(35,59)(39,60)(53,63)(58,62)

Polycyclic presentation:

2^10|b^2=g,c^2=gi,d^2=fgh,e^2=fj,b^a=bij,c^a=ci,c^b=cgj,d^a=di,d^b=dh,d^c=df,e^b=efh,e^c=egh,e^d=efhij,f^b=fi,f^c=fj,f^d=fi,g^d=gj,g^e=gij,h^b=hj,h^c=hij,h^d=hij,h^e=hi

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341133
maximal normal subgroup of	#72341133

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.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.7*F.8);;
gap> SetPower(coll, 5, F.6*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);