#72316587

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 7717 conjugacy classes of subgroups; only the 3397 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)(5,6,9,10)(7,11,12,8)(13,14,17,16)(15,19,20,18)(21,22)(23,25)(24,27)(26,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,25,27,26);(1,2,3,4)(6,10)(7,12)(13,15,17,20)(14,18,16,19)(21,23)(22,26)(24,28)(25,27);(6,10)(7,12)(13,14,17,16)(15,18,20,19);(5,8,9,11)(6,7,10,12)(13,16,17,14)(15,18,20,19);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,25)(24,27)(26,28)

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=ghi,d^2=ij,e^2=j,b^a=bj,c^a=cg,c^b=chj,d^a=di,d^b=di,e^b=ej,e^c=eij,f^c=fhj,f^d=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, 1, F.10);;
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> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*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, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);