#72302991

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 16242 conjugacy classes of subgroups; only the 31831 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)(13,14,17,20)(15,18,23,21)(16,19,25,28)(22,26,24,27);(5,6,8,10)(7,11,12,9)(13,15)(14,18)(16,24)(17,23)(19,27)(20,21)(22,25)(26,28);(2,4)(7,12)(9,11)(13,14,17,20)(15,18,23,21)(16,19,25,28)(22,26,24,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,12)(10,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,25)(27,28)

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=j,f^2=j,b^a=bh,d^b=dh,d^c=di,e^a=ej,e^b=ehj,e^c=ehj,f^b=fj,g^e=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303075
maximal normal subgroup of	#72303075

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