#453219

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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)(11,20,24,16)(12,25)(13,17)(14,28)(15,18)(19,29)(21,35,36,39)(22,31,37,30)(23,32,38,34)(26,40)(27,33);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,30)(18,20)(19,32)(25,39)(26,36)(27,37)(28,34)(29,38)(31,33)(35,40);(3,7)(5,8)(9,12,14,13,15,26,29,27)(10,17,19,25,18,33,28,40)(11,21,23,22,24,36,38,37)(16,30,32,39,20,31,34,35);(9,13,14,26,15,27,29,12)(10,17,28,40,18,33,19,25)(11,22,23,36,24,37,38,21)(16,31,34,39,20,30,32,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=f,d^2=ef,f^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,d^c=di

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, 360415);