#72312837

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 8100 conjugacy classes of subgroups; only the 31757 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,d^2=j,e^2=j,g^2=j,b^a=bi,c^b=ch,d^a=dj,d^b=dj,d^c=dhj,e^a=ej,e^b=ej,e^c=ej,e^d=ej,f^b=fi,g^a=gi,g^f=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312863, #72315226
maximal normal subgroup of	#72312863, #72315226

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