#483657

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	#121	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 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)(17,18,22,27,23,28,39,45,21,26,37,44,38,30,41,47)(19,29,32,46,33,35,25,43,31,24,40,42,48,20,34,36);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,24)(20,28)(21,31)(22,32)(23,33)(25,39)(26,29)(27,42)(30,35)(34,41)(36,45)(37,40)(38,48)(43,47)(44,46);(17,20,22,36,23,29,39,46,21,35,37,43,38,24,41,42)(18,25,27,31,28,40,45,48,26,34,44,19,30,32,47,33);(2,7)(6,15)(8,12)(9,16)(18,26)(19,31)(25,34)(27,44)(28,30)(32,40)(33,48)(45,47)

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,c^b=ci,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, 390853);