#508247

order	512
nilpotency class	4
isoclinism	isoclinic class of #508078
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×Q₈	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415443);