#6686375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25578	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,27,31)(23,25,32,28)(24,29,34,36)(26,30,35,33);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,28)(24,29)(25,31)(26,30)(27,32)(33,35)(34,36);(1,2,3,4)(21,24)(22,29)(23,26)(25,30)(27,34)(28,33)(31,36)(32,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,23,31,32)(24,33,34,30)(26,29,35,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26)(22,30)(23,24)(25,29)(27,35)(28,36)(31,33)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593571);