#6498303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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

1313 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(26,31)(27,36)(29,32)(33,38)(34,39)(35,40);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26,32,40,30,35,29,31)(27,33,28,38,36,39,37,34);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,27,30,36)(26,33,35,39)(28,29,37,32)(31,34,40,38);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,37)(26,34,35,38)(27,32,36,29)(31,39,40,33);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,36)(28,37)(33,39)(34,38)

Polycyclic presentation:

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

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, 6405499);