#6595069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,24,29,38)(20,32,33,41)(21,26,22,36)(25,35,40,34)(28,42,44,47)(30,39,31,46)(37,45,48,43);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,38)(20,28,33,44)(21,30,22,31)(25,37,40,48)(26,39,36,46)(32,42,41,47)(34,43,35,45);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,34,23,33,21,35)(18,25,26,32,27,40,36,41)(19,28,31,43,29,44,30,45)(24,37,39,42,38,48,46,47);(1,4,5,3)(2,8,6,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,33)(24,38)(25,40)(30,31)(32,41)(34,35)(39,46);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,33,35)(24,39,38,46)(25,41,40,32)(28,43,44,45)(37,47,48,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fg,d^2=f,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=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502265);