#453218

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
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

748 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)(11,16,23,34)(12,25)(13,18)(14,20)(15,30)(17,27)(19,29)(21,36,37,40)(22,31,28,32)(24,39,38,33)(26,35);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,21)(13,28)(14,23)(15,24)(17,31)(18,32)(19,33)(20,34)(22,27)(25,36)(26,37)(29,38)(30,39)(35,40);(3,7)(5,8)(9,12,15,27,14,26,29,13)(10,17,19,25,20,18,30,35)(11,21,24,22,23,37,38,28)(16,31,33,36,34,32,39,40);(9,13,29,26,14,27,15,12)(10,18,19,35,20,17,30,25)(11,22,38,21,23,28,24,37)(16,32,33,40,34,31,39,36)

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360414);