#126601

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

428 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,25,22,27,36,40,23,26,20,32,34,41,35,45)(19,28,29,37,31,42,43,47,30,24,33,38,44,39,46,48);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19)(18,24,26,28)(20,33)(21,29)(22,31)(23,30)(25,38,32,37)(27,39,41,42)(34,44)(35,46)(36,43)(40,48,45,47);(17,20,22,35,23,21,34,36)(18,25,27,40,26,32,41,45)(19,29,31,43,30,33,44,46)(24,37,39,47,28,38,42,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fgi,c^b=ci,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, 33797);