#410606

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 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)(3,5)(4,6)(7,8)(9,10,14,20,16,22,15,21)(11,23,26,37,27,40,33,17)(12,28,25,38,30,36,34,18)(13,31,24,39,32,19,29,35);(1,3,4,7)(2,5,6,8)(9,11,16,27)(10,17,22,37)(12,29,30,24)(13,25,32,34)(14,33,15,26)(18,35,38,39)(19,36,31,28)(20,40,21,23);(9,12,14,34,16,30,15,25)(10,18,21,28,22,38,20,36)(11,24,33,32,27,29,26,13)(17,35,23,19,37,39,40,31);(2,6)(5,8)(9,13,15,29,16,32,14,24)(10,19,21,39,22,31,20,35)(11,25,26,30,27,34,33,12)(17,36,23,18,37,28,40,38)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317802);