#6467958

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=hi,e^2=g,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, 6375154);