#72280903

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 8760 conjugacy classes of subgroups; only the 3605 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=hij,d^2=g,f^2=ij,b^a=bh,c^b=cij,d^a=dg,d^c=dg,e^b=ej,e^c=ei,f^a=fij,f^b=fij,f^c=fij,f^d=fi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281472
maximal normal subgroup of	#72281472

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