#6467957

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=h,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, 6375153);