#438972

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

737 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=ef,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, 346168);