#498228

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=dei,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405424);