#72232677

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

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

At least 6229 conjugacy classes of subgroups; only the 3677 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|c^2=g,e^2=j,f^2=hi,c^a=cg,c^b=ci,e^a=ej,e^d=ej,f^a=fj,f^b=fgh,f^c=fij,f^d=fgi,f^e=fgh

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710, #72326378, #72328873, …
maximal normal subgroup of	#72232710, #72326378, #72328873, …
maximal quotient of	#72333259, #72336446

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