#512015

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419211);