#445665

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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	#2535
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(3,6,4,8)(9,10,14,19,15,12,16,20)(11,21,24,27,25,22,26,17)(13,18,32,36,31,28,30,35)(23,34,39,33,38,37,29,40);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,17,12,27)(13,29,31,39)(14,26,16,24)(18,34,28,37)(19,22,20,21)(23,32,38,30)(33,36,40,35);(2,7)(6,8)(9,12,16,19,15,10,14,20)(11,22,26,27,25,21,24,17)(13,28,30,36,31,18,32,35)(23,34,29,40,38,37,39,33);(1,4,5,3)(2,8,7,6)(9,13,15,31)(10,18,12,28)(11,23,25,38)(14,32,16,30)(17,33,27,40)(19,36,20,35)(21,37,22,34)(24,39,26,29)

Polycyclic presentation:

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

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, 352861);