#415512

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

698 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,g^b=gi,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, 322708);