#72326381

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 14084 conjugacy classes of subgroups; only the 31817 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:

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

Polycyclic presentation:

2^10|b^a=bh,c^b=cj,f^b=fi,f^c=fj,f^d=fi,f^e=fhj,g^a=gj,g^b=gi,g^c=gj,g^d=gi,g^e=gh

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72326447, #72327136, #72328051

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