#72297945

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

22983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=hj,d^2=hj,e^2=hj,b^a=bh,d^a=dg,e^a=eghi,e^c=ehi,f^a=fghij,f^b=fh,f^c=fghij,f^d=fgij,f^e=fh

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297949
maximal normal subgroup of	#72297949
maximal quotient of	#72298624

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