#6555316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fghi,e^2=hi,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^b=ei,e^c=ei,e^d=ehi

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, 6462512);