#483753

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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

414 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18,22,28,24,27,41,46,23,29,40,31,21,37,39,47)(19,30,33,36,32,25,42,35,26,44,45,34,38,43,48,20);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,25)(20,31)(21,38)(22,33)(23,26)(24,32)(27,44)(28,35)(29,43)(30,37)(34,46)(36,47)(39,48)(40,45)(41,42);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(17,20,22,30,24,36,41,25,23,35,40,44,21,34,39,43)(18,26,28,45,27,38,46,48,29,19,31,33,37,32,47,42);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,21,23,24)(18,27,29,37)(19,32,26,38)(20,34,35,36)(22,39,40,41)(25,30,43,44)(28,46,31,47)(33,42,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390949);