#72321745

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄²
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂×D₄×C₄⋊C₄
prime radical	C₂×#575	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,41)(29,38)(30,45)(31,39)(33,46)(34,40)(37,44)(42,48)(43,47);(1,2,4,8,6,9,5,14)(3,10,11,15,13,16,12,7)(17,19,21,29,23,31,22,30)(18,24,26,38,27,39,36,45)(20,28,34,43,33,42,35,44)(25,37,32,41,40,47,46,48);(20,33)(25,40)(28,42)(32,46)(34,35)(37,47)(41,48)(43,44);(17,20,22,35,23,33,21,34)(18,25,26,32,27,40,36,46)(19,28,30,44,31,42,29,43)(24,37,38,41,39,47,45,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)

Polycyclic presentation:

2^10|b^2=f,d^2=g,e^2=h,f^2=i,g^2=ij,h^2=j,d^a=dg,d^c=dij,e^b=eh,g^a=gij,h^b=hj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321818, #72322246
maximal normal subgroup of	#72321818, #72322246

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