#6468449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(17,18,22,26)(19,27,29,42)(20,31,33,24)(21,35,37,25)(23,36,39,30)(28,44,32,43)(34,41,48,46)(38,45,40,47);(9,10,13,16)(11,15,12,14)(17,19,22,29)(18,23,26,39)(20,32,33,28)(21,36,37,30)(24,40,31,38)(25,42,35,27)(34,45,48,47)(41,43,46,44);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,38)(25,41)(26,31)(27,43)(29,32)(30,45)(35,46)(36,47)(37,48)(39,40)(42,44);(9,12,13,11)(10,15,16,14)(19,29)(23,39)(27,42)(28,32)(30,36)(38,40)(43,44)(45,47);(1,3,4,7)(2,6,5,8)(20,33)(24,31)(28,32)(34,48)(38,40)(41,46)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6375645);