#6468070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

1820 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)(17,18,22,26,23,27,37,43)(19,28,31,39,32,42,36,24)(20,33,34,48,21,25,35,41)(29,45,46,38,30,44,47,40);(2,5)(6,7)(10,14)(11,12)(17,19,23,32)(18,24,27,39)(20,30,21,29)(22,36,37,31)(25,40,33,38)(26,42,43,28)(34,46,35,47)(41,44,48,45);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,23,21)(18,25,27,33)(19,29,32,30)(22,34,37,35)(24,38,39,40)(26,41,43,48)(28,44,42,45)(31,46,36,47);(9,12,13,11)(10,16,14,15)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,37,34)(24,38,39,40)(26,41,43,48)(28,44,42,45)(31,47,36,46);(20,21)(25,33)(29,30)(34,35)(38,40)(41,48)(44,45)(46,47)

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^c=ei,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, 6375266);