#72328181

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9024 conjugacy classes of subgroups; only the 3097 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=i,d^2=i,b^a=bh,c^a=cj,c^b=cg,d^a=di,d^c=dj,e^a=ehj,e^b=ehj,e^d=ehj,f^a=fj,f^c=fj,f^e=fgh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328255
maximal normal subgroup of	#72328255

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