#483752

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	#466	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18,23,28,22,26,39,44,24,29,40,46,21,27,38,45)(19,30,33,36,31,43,47,37,34,42,48,35,32,25,41,20);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,25)(20,28)(21,32)(22,31)(23,33)(24,34)(26,30)(27,42)(29,43)(35,45)(36,44)(37,46)(38,41)(39,47)(40,48);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(17,20,23,30,22,36,39,43,24,37,40,42,21,35,38,25)(18,19,28,33,26,31,44,47,29,34,46,48,27,32,45,41);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,21,24,22)(18,26,29,27)(19,31,34,32)(20,35,37,36)(23,38,40,39)(25,42,43,30)(28,44,46,45)(33,47,48,41)

Polycyclic presentation:

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

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, 390948);