#72268221

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#2404
commutator, Frattini, cosocle	C₂⁶	C₂⁴
prime radical	C₂×#2403	C₂²
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

4445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,7,8,5)(13,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,32)(30,31);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23,25,26)(20,27)(21,30)(22,28)(24,32,29,31);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,24,25,29)(19,27)(21,22)(23,31,26,32)(28,30);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,31,26,32)

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=g,d^2=hj,b^a=be,c^a=cef,c^b=cg,d^a=dh,d^b=dj,d^c=dghij,e^b=ej,e^c=ei,f^b=fi,f^c=fij,f^d=fj,g^a=gj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268224
maximal normal subgroup of	#72268224

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