#6468057

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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)(17,18,23,37,22,27,21,26)(19,28,31,24,30,40,32,39)(20,25,34,42,36,44,35,43)(29,45,47,38,33,48,46,41);(2,5)(6,7)(10,14)(11,12)(17,19)(18,24)(20,33)(21,31)(22,30)(23,32)(25,41)(26,40)(27,39)(28,37)(29,36)(34,47)(35,46)(38,44)(42,48)(43,45);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,36)(18,25,27,44)(19,29,30,33)(21,35,23,34)(24,38,39,41)(26,43,37,42)(28,45,40,48)(31,47,32,46);(9,12,13,11)(10,16,14,15)(19,30)(24,39)(28,40)(29,33)(31,32)(38,41)(45,48)(46,47);(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,40,39,28)(25,42,44,43)(29,46,33,47)(38,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375253);