#72337210

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

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

306877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=j,g^2=j,b^a=bj,d^c=dj,g^a=gi,g^b=gh,g^c=gij,g^d=gh,g^e=ghi,g^f=gij

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72337222, #72337233, #72337254, …

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