#72340227

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	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,4)(5,13)(6,8)(7,14)(9,15)(10,11)(12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,37)(28,35)(29,36)(30,34)(31,38)(32,39)(33,40)(41,46)(42,45)(43,47)(44,48);(4,11)(5,14)(17,19)(18,23)(20,30)(21,28)(22,29)(24,37)(25,35)(26,36)(27,32)(31,42)(33,43)(34,39)(38,46)(40,47)(41,44)(45,48);(1,3)(2,4)(5,9,14,16)(6,10)(7,15,13,12)(8,11)(17,20,22,32)(18,24,26,39)(19,27)(21,31,33,44)(23,34)(25,38,40,48)(28,42)(29,30)(35,46)(36,37)(41,43)(45,47);(1,4)(2,3)(5,12,14,15)(6,11)(7,9,13,16)(8,10)(17,21,22,33)(18,25,26,40)(19,28)(20,31,32,44)(23,35)(24,38,39,48)(27,41)(29,43)(30,42)(34,45)(36,47)(37,46);(1,5)(2,7)(3,9)(4,12)(6,14)(8,13)(10,16)(11,15)(18,26)(21,33)(23,36)(24,39)(34,37)(35,47)(38,48)(41,42)

Polycyclic presentation:

2^10|c^2=gij,d^2=gij,b^a=bf,c^a=cg,c^b=cfh,d^b=dhi,d^c=dg,e^a=ei,e^b=ehj,e^c=ehj,e^d=efgi,f^c=fi,f^d=fj,f^e=fj,g^b=gij,g^e=gj,h^a=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340239, #72340295, #72340305, …
maximal normal subgroup of	#72340239, #72340295, #72340305, …

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