#72263117

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 10604 conjugacy classes of subgroups; only the 3625 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:

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

Polycyclic presentation:

2^10|c^2=i,d^2=j,f^2=i,b^a=bh,c^a=ch,c^b=ci,d^b=dhj,d^c=dij,e^a=ej,e^b=eg,f^a=fh,f^b=fi,f^d=fgij,f^e=fj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263129
maximal normal subgroup of	#72263129

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