#494122

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401318);