#72328902

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 10622 conjugacy classes of subgroups; only the 3321 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|c^2=hi,e^2=h,f^2=hj,b^a=bi,c^a=cij,c^b=ch,d^a=dij,d^c=dhi,e^a=eij,e^b=eh,e^c=ei,e^d=eh,f^d=fhj,f^e=fg

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329012
maximal normal subgroup of	#72329012
maximal quotient of	#72331262

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