#72263085

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 22956 conjugacy classes of subgroups; only the 31741 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:

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

Polycyclic presentation:

2^10|b^2=h,g^2=i,b^a=bh,c^a=ch,c^b=ch,e^a=ej,e^b=ehi,f^b=fh,f^c=fi,f^d=fj,g^a=gh,g^c=gi,g^d=ghij,g^e=gj,g^f=gi

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72263129

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