#6569332

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2657	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(9,10,15,14)(11,12,13,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,30)(25,36)(29,42)(31,41)(32,40)(34,39)(35,43)(37,48)(44,45)(46,47);(2,5)(6,8)(9,11,15,13)(10,16,14,12)(17,19,22,30,23,32,21,31)(18,24,26,28,27,41,38,40)(20,34,36,46,37,29,35,45)(25,42,33,47,43,39,48,44);(1,3,4,7)(2,6,5,8)(9,12,15,16)(10,11,14,13)(17,20,21,35,23,37,22,36)(18,25,38,48,27,43,26,33)(19,29,31,46,32,34,30,45)(24,39,40,47,41,42,28,44);(9,13,15,11)(10,16,14,12)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,33,43,48)(29,45,34,46)(39,47,42,44);(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,26,27,38)(19,31,32,30)(20,35,37,36)(24,28,41,40)(25,33,43,48)(29,46,34,45)(39,44,42,47)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476528);