#6534572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 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)(3,6)(4,5)(7,8)(17,18,23,28)(19,29,31,40)(20,34,35,42)(21,26,22,27)(24,33,39,32)(25,37,41,36)(30,44,47,38)(43,46,48,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,28,39)(20,30,35,47)(21,33,22,32)(25,38,41,44)(26,29,27,40)(34,48,42,43)(36,46,37,45);(10,13)(14,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,42)(27,34)(28,41)(29,43)(31,47)(32,45)(33,46)(39,44)(40,48);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,35,37)(24,40,39,29)(25,42,41,34)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441768);