#408890

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1021 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)(4,7)(5,6)(9,10,16,11,14,19,15,22)(12,20,29,32,24,31,25,17)(13,21,28,33,26,35,27,18)(23,36,39,40,37,30,38,34);(2,6)(7,8)(9,11,15,19,14,22,16,10)(12,17,25,20,24,32,29,31)(13,21,27,33,26,35,28,18)(23,30,38,34,37,36,39,40);(1,3,5,4)(2,7,6,8)(9,12,15,25,14,24,16,29)(10,17,11,20,19,32,22,31)(13,23,27,38,26,37,28,39)(18,30,21,34,33,36,35,40);(1,4,5,3)(2,7,6,8)(9,13,16,28,14,26,15,27)(10,18,22,35,19,33,11,21)(12,23,29,39,24,37,25,38)(17,30,31,40,32,36,20,34)

Polycyclic presentation:

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

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, 316086);