#72305137

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

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

45965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=g,d^2=gj,e^2=j,f^2=gj,b^a=bg,d^b=dj,e^b=ej,f^a=fgh,f^b=fi,f^c=fhj,f^e=fgj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305143
maximal normal subgroup of	#72305143

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