#484136

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391332);