#72304070

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 7901 conjugacy classes of subgroups; only the 2985 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)(10,11)(12,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,30)(28,29)(31,32);(3,7)(5,8)(10,11)(14,15)(25,26)(27,31)(28,29)(30,32);(9,10,13,11)(12,15,16,14)(17,19,21,22)(18,20,23,24)(26,29)(30,31);(17,20,21,24)(18,22,23,19)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,5,6,8)(9,11,13,10)(12,14,16,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,2)(3,8)(4,6)(5,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,20)(21,22)(23,24)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^10|a^2=g,c^2=gi,d^2=j,e^2=ij,c^a=ch,c^b=chj,d^c=dghij,e^a=egi,e^b=eghj,e^c=ej,e^d=ej,f^a=fi,f^c=fh,f^e=fij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304145
maximal normal subgroup of	#72304145

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