#6464884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

1392 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6372080);