#72316564

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 8366 conjugacy classes of subgroups; only the 3409 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=ghi,d^2=ij,c^a=cg,c^b=chj,d^b=dj,d^c=dj,e^c=ehi,e^d=ei,f^a=fj,f^d=fij,f^e=fj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316598
maximal normal subgroup of	#72316598

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