#72320341

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#19125	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=ehij,d^2=egi,e^2=i,f^2=h,g^2=i,h^2=j,c^b=cf,d^a=dg,d^b=di,d^c=dhj,e^c=ej,f^b=fh,f^c=fh,f^d=fj,g^a=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320455, #72320885, #72321935
maximal normal subgroup of	#72320455, #72320885, #72321935

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