#127043

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fi,d^b=di,e^b=eg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34239);