#6499324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26,30,34,29,36,28,35)(27,33,37,31,39,32,38,40);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,29,39)(26,31,36,40)(28,37,30,38)(32,34,33,35);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,20)(19,24)(25,27)(26,32)(28,38)(29,39)(30,37)(31,35)(33,36)(34,40);(11,18)(14,17)(19,24)(22,23)(25,27)(26,33)(28,38)(29,39)(30,37)(31,34)(32,36)(35,40);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,28,29,30)(26,34,36,35)(27,37,39,38)(31,33,40,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

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, 6406520);