#72342314

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 9926 conjugacy classes of subgroups; only the 31825 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:

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

Polycyclic presentation:

2^10|d^2=j,f^2=hj,g^2=j,d^a=dh,d^c=dj,e^b=ehj,e^d=ej,f^b=fhj,f^c=fhj,f^d=fj,g^a=gi,g^e=gij,g^f=gij

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342349
maximal normal subgroup of	#72342349

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