#6592725

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1713 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:

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

Polycyclic presentation:

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

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, 6499921);