#72256602

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 13100 conjugacy classes of subgroups; only the 31809 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)(15,21)(16,20)(17,23)(18,19)(22,25)(24,27)(26,28);(2,4)(13,15,18,23)(14,17,19,21)(16,22,24,28)(20,26,27,25);(1,2,3,4)(5,6,8,10)(7,9,12,11);(5,6,8,10)(7,11,12,9)(14,19)(16,24)(17,21)(22,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,25)(23,28);(5,7)(6,9)(8,12)(10,11)(13,17)(14,15)(16,25)(18,21)(19,23)(20,28)(22,27)(24,26);(13,14)(15,17)(16,20)(18,19)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^10|b^2=j,c^2=hi,d^2=i,b^a=bj,c^a=ci,c^b=ch,d^a=dij,e^c=ei,e^d=eij,f^a=fj,f^b=fj,f^d=fij,f^e=fj,g^d=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256608, #72256702
maximal normal subgroup of	#72256608, #72256702
maximal quotient of	#72256795, #72263489

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