#483537

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,26,21,28,38,45,23,27,35,30,39,25,42,44)(19,29,33,37,34,41,47,36,31,43,46,48,32,24,40,20);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,31)(18,24,27,41)(20,30,36,26)(21,34,39,32)(22,33,35,46)(25,43,28,29)(37,44,48,45)(38,47,42,40);(17,20,35,43,21,37,42,24,23,36,22,29,39,48,38,41)(18,19,30,46,28,34,44,40,27,31,26,33,25,32,45,47);(19,31)(20,36)(24,41)(29,43)(32,34)(33,46)(37,48)(40,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,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, 390733);