#498734

order	512
nilpotency class	3
isoclinism	isoclinic class of #384166
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2377
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23333
prime radical	#3054	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,12,5,11,20,27,7,9,22,29,21,28,4,10)(3,13,17,31,18,8,19,25,14,26,15,30,16,23,32,24);(1,3)(2,8)(4,19)(5,18)(6,15)(7,14)(9,23)(10,30)(11,26)(12,24)(13,28)(16,21)(17,22)(20,32)(25,29)(27,31);(2,9)(3,14)(4,20)(6,22)(8,23)(11,28)(13,26)(16,18);(1,4,5,6,7,20,21,22)(2,10,11,12,9,27,28,29)(3,15,16,19,14,17,18,32)(8,24,13,30,23,25,26,31)

Polycyclic presentation:

2^9|a^2=g,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cefh,d^b=de,e^b=ei,f^b=fi,g^b=gi,d^c=di,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405930);