#72287850

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7624082	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2185 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=gij,e^2=f,f^2=h,g^2=ij,i^2=j,c^a=ch,c^b=chj,d^a=dj,d^b=dg,e^a=eh,e^b=ej,e^d=egj,f^d=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318539, #72319058, #72319103
maximal normal subgroup of	#72318539, #72319058, #72319103

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