#453220

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	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

755 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)(9,10,14,32,15,20,16,21)(11,22,33,37,24,17,23,36)(12,25,27,18,28,40,29,35)(13,30,34,19,26,39,31,38);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,17,20,22)(12,26,28,13)(14,23,16,33)(18,38,35,19)(21,36,32,37)(25,30,40,39)(27,34,29,31);(3,7)(5,8)(9,12,14,27,15,28,16,29)(10,18,21,25,20,35,32,40)(11,13,23,31,24,26,33,34)(17,19,36,39,22,38,37,30);(9,13,14,31,15,26,16,34)(10,19,32,30,20,38,21,39)(11,12,23,27,24,28,33,29)(17,35,37,25,22,18,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 360416);