#452629

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 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)(3,5,7,8)(9,10)(11,16,23,31)(12,25)(13,29,24,34)(14,20)(15,19)(17,28)(18,21)(22,36,37,40)(26,39,38,30)(27,32)(33,35);(1,3)(2,5)(4,7)(6,8)(9,11,21,13,14,23,15,24)(10,16,18,29,20,31,19,34)(12,22,35,26,27,37,28,38)(17,30,25,36,33,39,32,40);(3,7)(5,8)(9,12,15,28,14,27,21,35)(10,17,18,25,20,33,19,32)(11,22,24,38,23,37,13,26)(16,30,29,36,31,39,34,40);(9,13,21,23,14,24,15,11)(10,16,18,29,20,31,19,34)(12,26,35,37,27,38,28,22)(17,30,25,36,33,39,32,40)

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359825);