#72335772

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 6703 conjugacy classes of subgroups; only the 3821 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:

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

Polycyclic presentation:

2^10|e^2=i,f^2=gj,b^a=bg,c^b=ci,e^a=ej,e^c=egh,e^d=ehij,f^a=fj,f^c=fhij,f^d=fhij

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335781
maximal normal subgroup of	#72335781
maximal quotient of	#72336149

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