#438718

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

809 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)(9,10,15,20)(11,22,26,40)(12,28,24,17)(13,31,33,18)(14,21,34,19)(16,27,36,25)(23,37,32,39)(29,38,30,35);(9,11,14,27,15,26,34,25)(10,16,21,22,20,36,19,40)(12,13,30,32,24,33,29,23)(17,37,38,31,28,39,35,18);(1,3,4,8)(2,5,6,7)(9,12,15,24)(10,17,20,28)(11,23,26,32)(13,27,33,25)(14,29,34,30)(16,18,36,31)(19,38,21,35)(22,39,40,37);(9,13,14,32,15,33,34,23)(10,18,19,39,20,31,21,37)(11,24,25,30,26,12,27,29)(16,35,22,17,36,38,40,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 345914);