#483727

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,27,21,29,37,46,23,28,39,31,38,26,44,45)(19,30,33,36,34,41,47,35,25,42,43,48,32,24,40,20);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,25)(18,24,28,41)(20,31,35,27)(21,34,38,32)(22,33,39,43)(26,42,29,30)(36,45,48,46)(37,47,44,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,22,30,21,36,37,41,23,35,39,42,38,48,44,24)(18,25,27,43,29,32,46,40,28,19,31,33,26,34,45,47);(2,7)(6,15)(8,12)(9,16)(19,25)(20,35)(24,41)(30,42)(32,34)(33,43)(36,48)(40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390923);