#6467050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,31,23)(20,24,34,40)(22,26,29,36)(28,41,43,35)(30,39,33,37)(32,38,44,47)(42,48,45,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(19,28)(20,32)(22,29)(23,35)(24,38)(26,36)(27,41)(30,33)(31,43)(34,44)(37,39)(40,47);(9,12,13,11)(10,14,15,16)(17,19,29,43,21,31,22,28)(18,23,36,41,25,27,26,35)(20,33,44,42,34,30,32,45)(24,39,47,46,40,37,38,48);(17,20,29,44,21,34,22,32)(18,24,36,47,25,40,26,38)(19,30,43,45,31,33,28,42)(23,37,41,48,27,39,35,46);(1,3,4,7)(2,6,5,8)(18,25)(20,34)(23,27)(26,36)(30,33)(32,44)(35,41)(42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6374246);