#72256675

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 14026 conjugacy classes of subgroups; only the 31811 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=j,d^2=hi,e^2=i,b^a=bj,d^a=di,d^b=dh,d^c=dh,e^a=eij,e^d=ej,f^d=fi,f^e=fi,g^a=gj,g^b=gj,g^c=gj,g^e=gij,g^f=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256702
maximal normal subgroup of	#72256702

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