#6569309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,38)(31,39)(32,40)(35,42)(37,41)(43,46)(44,47)(45,48);(2,5)(6,8)(9,11,15,13)(10,16,14,12)(17,19,22,33,23,32,21,31)(18,24,26,29,28,39,27,40)(20,30,35,44,37,45,36,46)(25,38,41,43,42,48,34,47);(1,3,4,7)(2,6,5,8)(9,12,15,16)(10,11,14,13)(17,20,21,36,23,37,22,35)(18,25,27,34,28,42,26,41)(19,30,31,46,32,45,33,44)(24,38,40,47,39,48,29,43);(9,13,15,11)(10,16,14,12)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,29,39,40)(25,34,42,41)(30,44,45,46)(38,47,48,43);(9,14,15,10)(11,12,13,16)(19,32)(24,39)(29,40)(30,45)(31,33)(38,48)(43,47)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476505);