#6458266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^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,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, 6365462);