#438973

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,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^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, 346169);