#6534578

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,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, 6441774);