#72339910

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|c^2=j,d^2=hi,b^a=bg,c^b=ci,d^a=dgh,d^b=di,d^c=dij,e^b=egj,e^d=egj,f^b=fhj,f^d=fhj

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339917, #72339952
maximal normal subgroup of	#72339917, #72339952
maximal quotient of	#72339934, #72339976, #72340046

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