#425444

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332640);