#6466468

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 6373664);