#498034

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405230);