#72330752

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 22273 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) = 64

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^10|b^a=bj,c^b=ch,f^b=fh,f^c=fi,f^d=fhi,f^e=fh,g^a=gi,g^c=gi,g^d=gh

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72330942, #72331604, #72331723, …

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