#72326920

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 14606 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:

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

Polycyclic presentation:

2^10|f^2=hj,g^2=j,b^a=bj,c^b=ch,f^a=fi,f^c=fj,f^d=fi,f^e=fhj,g^a=ghi,g^c=gj,g^d=gi,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72327062, #72328000, #72329191, …

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