#72239195

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄⁴ · Schur multiplier (H₂) = C₄⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₄	C₄⁴
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#353107
prime radical	C₂⁴	#568
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,c^2=g,d^2=h,f^2=i,i^2=j,c^a=ci,d^b=dij,e^c=ej,f^d=fj,g^a=gj,h^b=hj

Conjugacy classes & character table

274 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239203
maximal normal subgroup of	#72239203

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