#6468439

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

1500 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)(20,24,32,40)(21,34,36,25)(23,35,38,30)(28,43,31,45)(33,48,47,41)(37,46,39,44);(9,10,13,16)(11,15,12,14)(17,19,22,29)(18,23,26,38)(20,31,32,28)(21,35,36,30)(24,39,40,37)(25,42,34,27)(33,44,47,46)(41,43,48,45);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,22,32)(18,24,26,40)(19,28,29,31)(21,33,36,47)(23,37,38,39)(25,41,34,48)(27,43,42,45)(30,44,35,46);(9,12,13,11)(10,15,16,14)(19,29)(23,38)(27,42)(28,31)(30,35)(37,39)(43,45)(44,46);(1,3,4,7)(2,6,5,8)(20,32)(24,40)(28,31)(33,47)(37,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=g,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, 6375635);