#72337572

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
prime radical	#251827	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2782 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(29,34)(32,45)(37,41)(38,44);(1,3)(2,6)(4,11,12,9)(5,10)(7,13,15,16)(8,14)(17,18)(19,27,30,23)(20,31,33,24)(21,25)(22,26)(28,36)(29,41,34,37)(32,44,45,38)(35,39)(40,43)(42,47)(46,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,15)(10,11)(14,16)(17,19,22,30)(18,23,26,27)(20,28,33,43)(21,34,35,29)(24,36,31,40)(25,37,39,41)(32,46,45,42)(38,47,44,48);(2,8)(7,15)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,38)(26,31)(27,40)(29,42)(30,43)(34,46)(35,45)(37,47)(39,44)(41,48);(7,15)(13,16)(17,21)(18,25)(19,29)(20,32)(22,35)(23,37)(24,38)(26,39)(27,41)(28,42)(30,34)(31,44)(33,45)(36,47)(40,48)(43,46)

Polycyclic presentation:

2^10|b^2=h,c^2=j,c^a=cf,c^b=cfg,d^a=dfi,d^b=dg,e^a=eh,e^c=egj,f^b=fi,f^e=fj,g^a=gi,g^b=gj,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337579, #72340239, #72340248, …
maximal normal subgroup of	#72337579, #72340239, #72340248, …

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