#72342299

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 9080 conjugacy classes of subgroups; only the 31813 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:

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

Polycyclic presentation:

2^10|d^2=j,f^2=hj,d^a=dh,d^c=dj,e^b=ehj,e^d=ei,f^b=fhj,f^c=fhj,f^d=fi,g^a=gi,g^d=gj

Conjugacy classes & character table

184 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> 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.9);;
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.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);