#415559

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=egh,b^2=ef,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=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, 322755);