#6468066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,36,22,27,21,26)(19,28,35,24,29,43,31,38)(20,25,37,48,33,41,34,42)(30,44,47,39,32,46,45,40);(2,5)(6,7)(10,14)(11,12)(17,19,22,29)(18,24,27,38)(20,32,33,30)(21,35,23,31)(25,40,41,39)(26,43,36,28)(34,45,37,47)(42,44,48,46);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(19,29)(24,38)(28,43)(30,32)(31,35)(39,40)(44,46)(45,47);(9,12,13,11)(10,16,14,15)(17,20,22,33)(18,25,27,41)(19,30,29,32)(21,34,23,37)(24,39,38,40)(26,42,36,48)(28,44,43,46)(31,45,35,47);(20,33)(25,41)(30,32)(34,37)(39,40)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375262);