#410731

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
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	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317927);