#72300780

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

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 9500 conjugacy classes of subgroups; only the 3017 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=h,c^2=i,e^2=gj,b^a=bg,c^b=ci,d^c=dij,e^a=eghij,e^b=egj,e^d=egj,f^b=fi,f^c=fi,f^d=fh,f^e=fi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300808
maximal normal subgroup of	#72300808

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