#439576

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1108 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)(9,10,16,34,15,21,14,20)(11,22,33,17,26,38,25,37)(12,27,30,36,29,18,24,40)(13,31,28,39,32,35,23,19);(1,3,5,4)(2,6,7,8)(9,11,14,33,15,26,16,25)(10,17,20,38,21,37,34,22)(12,28,30,13,29,23,24,32)(18,31,40,19,27,35,36,39);(1,4,5,3)(2,6,7,8)(9,12,14,24,15,29,16,30)(10,18,20,36,21,27,34,40)(11,23,25,32,26,28,33,13)(17,35,22,39,37,31,38,19);(9,13,14,23,15,32,16,28)(10,19,20,35,21,39,34,31)(11,24,25,29,26,30,33,12)(17,36,22,27,37,40,38,18)

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=e,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^b=gi,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, 346772);