#483491

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,e^b=eg,d^c=di

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, 390687);