#415511

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

708 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)(9,10,14,21,16,20,15,31)(11,22,25,33,12,17,24,18)(13,27,28,35,30,40,29,19)(23,34,38,36,26,39,37,32);(2,5)(6,8)(9,11,14,24,16,12,15,25)(10,17,21,33,20,22,31,18)(13,23,28,37,30,26,29,38)(19,32,27,39,35,36,40,34);(1,3,4,7)(2,6,5,8)(9,12)(10,18,20,33)(11,16)(13,26)(14,25)(15,24)(17,31,22,21)(19,34,35,39)(23,30)(27,32,40,36)(28,38)(29,37);(9,13,15,29,16,30,14,28)(10,19,21,27,20,35,31,40)(11,23,25,38,12,26,24,37)(17,32,33,39,22,36,18,34)

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,g^b=gi,g^c=gi

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, 322707);