#6462309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

1469 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)(17,18,21,26,22,27,35,43)(19,28,30,38,31,42,34,23)(20,25,33,41,32,24,39,40)(29,45,47,36,46,44,48,37);(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,23,27,38)(20,29,32,46)(21,34,35,30)(24,36,25,37)(26,42,43,28)(33,48,39,47)(40,45,41,44);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,27)(26,40)(28,44)(30,47)(31,46)(34,48)(35,39)(37,38)(41,43)(42,45);(10,14)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,27)(26,41)(28,45)(30,47)(31,46)(34,48)(35,39)(36,38)(40,43)(42,44);(1,3,4,7)(2,6,5,8)(20,32)(24,25)(29,46)(33,39)(36,37)(40,41)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=gi,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, 6369505);