#72321387

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂²×D₄⋊C₈
prime radical	#61427	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ C₈×#131 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=e,e^2=g,f^2=hj,g^2=i,h^2=j,b^a=bf,c^a=cj,c^b=cj,d^c=dij,e^a=eh,f^a=fh,f^b=fj,g^a=gj,h^a=hj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321428, #72321481, #72321655
maximal normal subgroup of	#72321428, #72321481, #72321655
maximal quotient of	#72323188

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