#446298

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fg,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, 353494);