#6625102

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(17,18,22,25)(19,27,29,41)(20,31,33,42)(21,34,37,26)(23,35,39,30)(24,36,40,32)(28,38,44,47)(43,46,48,45);(2,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,28)(21,35)(22,29)(24,38)(25,39)(26,41)(27,34)(30,37)(31,43)(32,46)(33,44)(36,45)(40,47)(42,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,33)(18,24,25,40)(19,28,29,44)(21,36,37,32)(23,38,39,47)(26,42,34,31)(27,43,41,48)(30,46,35,45);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(18,25)(23,39)(24,40)(26,34)(27,41)(31,42)(38,47)(43,48);(9,12,13,11)(10,16,14,15)(19,29)(23,39)(27,41)(28,44)(30,35)(38,47)(43,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6532298);