#72327652

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 42969 conjugacy classes of subgroups; only the 31745 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:

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

Polycyclic presentation:

2^10|f^2=ij,g^2=hj,b^a=bh,c^b=chi,f^a=fhj,f^d=fhj,f^e=fij,g^a=gij,g^d=gj,g^e=ghj,g^f=gh

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72327934, #72329191, #72329601

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