#126707

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=chi

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, 33903);