#495936

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×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×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

596 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)(3,6)(4,5)(7,8)(9,10,15,19,13,16,28,20)(11,21,24,34,23,38,25,40)(12,18,27,36,26,32,14,29)(17,33,35,39,31,30,37,22);(2,5)(6,8)(10,16)(17,31)(18,32)(19,20)(21,38)(29,36)(34,40)(35,37);(1,3,4,7)(2,6,5,8)(9,11,13,23)(10,17,16,31)(12,22,26,39)(14,30,27,33)(15,24,28,25)(18,34,32,40)(19,35,20,37)(21,36,38,29);(9,12,15,27,13,26,28,14)(10,18,19,36,16,32,20,29)(11,22,24,33,23,39,25,30)(17,34,35,38,31,40,37,21)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403132);