#451682

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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)(9,10,16,20,14,21,15,32)(11,17,25,39,24,37,27,36)(12,26,23,35,28,40,29,18)(13,30,31,38,22,19,33,34);(1,3)(2,5)(4,7)(6,8)(9,11,16,27,14,24,15,25)(10,17,20,36,21,37,32,39)(12,22,23,31,28,13,29,33)(18,34,26,19,35,38,40,30);(3,7)(5,8)(9,12,15,29,14,28,16,23)(10,18,20,26,21,35,32,40)(11,22,25,33,24,13,27,31)(17,34,36,19,37,38,39,30);(9,13,14,22)(10,19)(11,23,24,29)(12,27,28,25)(15,31,16,33)(17,35)(18,37)(20,38)(21,30)(26,39)(32,34)(36,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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, 358878);