#410776

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

712 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)(3,8)(4,6)(5,7)(9,10,14,19,15,20,13,17)(11,21,22,32,24,16,25,31)(12,26,28,18,29,33,27,34)(23,37,36,35,38,40,39,30);(9,11,14,22,15,24,13,25)(10,16,17,32,20,21,19,31)(12,23,28,36,29,38,27,39)(18,30,26,40,34,35,33,37);(1,3,5,4)(2,6,7,8)(10,17,20,19)(11,22,24,25)(18,26,34,33)(23,36,38,39);(1,4,5,3)(2,6,7,8)(9,12,13,27,15,29,14,28)(10,18,19,33,20,34,17,26)(11,23,25,39,24,38,22,36)(16,30,31,37,21,35,32,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=fg,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=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, 317972);