#6555178

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 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)(25,26,31,29)(27,36,39,28)(30,34,40,35)(32,33,37,38);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38,31,39,40,33)(26,32,35,36,29,37,34,28);(1,3,5,4)(2,7,8,6)(10,15)(11,17)(16,22)(18,23)(25,28,40,37,31,36,30,32)(26,33,34,39,29,38,35,27);(1,4,5,3)(2,6,8,7)(12,19)(16,22)(18,23)(20,24)(25,29,31,26)(27,37,39,32)(28,33,36,38)(30,34,40,35);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,30,31,40)(26,34,29,35)(27,33,39,38)(28,32,36,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgh,e^2=hi,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=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, 6462374);