#6467954

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,34)(29,47)(30,40)(31,41)(32,42)(35,45)(36,39)(37,44)(38,46)(43,48);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,30)(21,36)(22,31)(23,38)(25,43)(26,39)(27,41)(28,46)(29,35)(32,37)(33,40)(34,48)(42,44)(45,47);(10,13)(14,16)(17,20,23,35,22,34,37,21)(18,25,28,26,27,33,44,45)(19,29,32,30,31,36,38,48)(24,39,42,43,41,47,46,40);(17,21,37,34,22,35,23,20)(18,26,44,25,27,45,28,33)(19,30,38,29,31,48,32,36)(24,40,46,47,41,43,42,39);(9,11,12,15)(10,14,13,16)(18,27)(24,41)(25,33)(26,45)(28,44)(39,47)(40,43)(42,46)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=hi,e^2=gi,h^2=i,b^a=bf,c^a=cg,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, 6375150);