#410779

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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)(9,10,13,17,14,19,28,18)(11,20,22,15,23,30,40,31)(12,24,25,33,27,16,26,34)(21,36,37,29,39,35,38,32);(9,11,13,22,14,23,28,40)(10,15,18,20,19,31,17,30)(12,21,26,38,27,39,25,37)(16,32,34,36,24,29,33,35);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,21)(13,26)(14,27)(15,29)(17,33)(18,34)(19,24)(20,35)(22,38)(23,39)(25,28)(30,36)(31,32)(37,40);(1,4,5,3)(2,6,7,8)(10,17,19,18)(15,30,31,20)(16,33,24,34)(29,36,32,35)

Polycyclic presentation:

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

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, 317975);