#438437

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 345633);