#484038

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	#2186	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=di,e^a=eg,g^a=gi,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, 391234);