#72233491

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

471923 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|b^2=j,d^2=j,b^a=bj,c^b=cj,d^a=dj,g^e=gj,g^f=gi,h^e=hi

Conjugacy classes & character table

277 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260648
maximal normal subgroup of	#72260648
maximal quotient of	#72233731, #72234844, #72260847

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