#483536

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
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	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390732);