#6620015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,28,31,24)(20,32,34,42)(21,27,23,36)(25,37,41,33)(29,44,46,48)(30,40,35,39)(38,47,43,45);(2,6)(7,8)(10,14)(11,12)(17,19)(18,24)(20,29)(21,35)(22,31)(23,30)(25,38)(26,28)(27,40)(32,48)(33,47)(34,46)(36,39)(37,45)(41,43)(42,44);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,34)(18,25,26,41)(19,29,31,46)(21,33,23,37)(24,38,28,43)(27,42,36,32)(30,45,35,47)(39,48,40,44);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(18,26)(24,28)(25,41)(27,36)(32,42)(38,43)(39,40)(44,48);(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,27,26,36)(19,30,31,35)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,45,46,47)(38,48,43,44)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527211);