#72256689

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 8289 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)(7,11)(8,9)(10,12)(15,22)(16,20)(23,25)(24,28);(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,28)(26,27);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,20)(21,27)(24,28);(1,2,3,4)(6,9)(10,11);(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25);(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,28)(22,25)

Polycyclic presentation:

2^10|d^2=hi,e^2=i,f^2=j,c^b=cj,d^a=di,d^b=dh,d^c=dh,e^a=eij,e^b=eij,f^a=fj,f^c=fj,f^d=fi,g^e=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, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*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, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);