#72233739

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

At least 6010 conjugacy classes of subgroups; only the 417519 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=i,d^2=i,b^a=bi,c^b=ci,d^a=di,d^c=dij,g^e=gj,g^f=gi,h^e=hij,h^f=hi

Conjugacy classes & character table

277 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72234468, #72234898, #72237405, …

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