#72232642

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 10670 conjugacy classes of subgroups; only the 3337 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)(13,14)(15,20)(16,18)(17,19)(22,23)(26,27);(1,2)(3,4)(15,17)(19,20)(24,28)(26,27);(2,4)(8,12)(10,11)(13,15,18,17)(14,19,16,20)(24,28)(26,27);(6,7)(10,11)(14,16)(19,20)(21,22,25,23)(24,26,28,27);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,17,19)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,19,16,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^10|c^2=g,d^2=h,e^2=j,f^2=g,c^a=cg,c^b=ci,d^a=dj,e^a=ej,e^d=ehj,f^b=fgh,f^c=fij,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710
maximal normal subgroup of	#72232710

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