#438436

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,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, 345632);