#6466665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,33)(21,36)(22,27)(23,28)(25,34)(26,35)(29,44)(30,47)(31,42)(32,43)(37,40)(38,45)(39,46)(41,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,37)(22,31)(23,39)(25,41)(26,44)(27,42)(28,46)(29,35)(32,38)(33,47)(34,48)(36,40)(43,45);(9,12,13,11)(10,14,15,16)(17,20,23,35,22,34,38,21)(18,25,28,36,27,33,45,26)(19,29,32,30,31,37,39,48)(24,40,43,41,42,44,46,47);(17,21,38,34,22,35,23,20)(18,26,45,33,27,36,28,25)(19,30,39,29,31,48,32,37)(24,41,46,40,42,47,43,44);(1,3,4,7)(2,6,5,8)(18,27)(24,42)(25,33)(26,36)(28,45)(40,44)(41,47)(43,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373861);