#72266572

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 7755 conjugacy classes of subgroups; only the 2973 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,22)(21,23)(25,26)(27,32)(28,30)(29,31);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23);(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,23,24)(25,27,29,28)(26,30,31,32);(11,16)(13,15)(25,26)(27,32)(28,30)(29,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,22,23,24)(25,28,29,27)(26,30,31,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31)

Polycyclic presentation:

2^10|a^2=i,c^2=j,e^2=j,b^a=bh,c^a=cij,d^c=dg,e^a=eh,e^b=egj,f^a=fh,f^b=fh,f^d=fi,f^e=fj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266583, #72266584
maximal normal subgroup of	#72266583, #72266584
maximal quotient of	#72337261

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