#6592055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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

1706 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6499251);