#410607

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
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

778 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)(11,23,26,36,27,40,33,17)(12,28,30,37,31,35,25,18)(13,19,24,38,32,39,29,34);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,29)(13,25)(14,33)(15,26)(16,27)(18,34)(19,35)(20,40)(21,23)(22,36)(24,31)(28,39)(30,32)(37,38);(9,12,14,25,16,31,15,30)(10,18,21,28,22,37,20,35)(11,24,33,32,27,29,26,13)(17,34,23,39,36,38,40,19);(2,6)(5,8)(9,13,15,29,16,32,14,24)(10,19,21,34,22,39,20,38)(11,25,26,12,27,30,33,31)(17,35,23,18,36,28,40,37)

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,d^c=di,e^c=ei

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, 317803);