#483763

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

406 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=g,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, 390959);