#6468473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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)(17,18,22,24)(19,27,28,42)(20,25,32,39)(21,34,36,26)(23,35,38,30)(29,43,31,45)(33,48,47,41)(37,46,40,44);(9,10,13,16)(11,15,12,14)(17,19,22,28)(18,23,24,38)(20,31,32,29)(21,35,36,30)(25,40,39,37)(26,42,34,27)(33,44,47,46)(41,43,48,45);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(18,24)(19,28)(25,39)(26,34)(29,31)(30,35)(41,48)(44,46);(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,39)(26,41)(27,43)(28,31)(30,44)(34,48)(35,46)(36,47)(38,40)(42,45);(1,3,4,7)(2,6,5,8)(20,32)(25,39)(29,31)(33,47)(37,40)(41,48)(43,45)(44,46)

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^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, 6375669);