#410732

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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

648 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)(9,10,16,20,15,21,14,33)(11,22,31,35,25,37,24,17)(12,26,34,40,29,18,28,36)(13,30,23,38,32,39,27,19);(2,5)(6,8)(9,11,14,24,15,25,16,31)(10,17,20,22,21,35,33,37)(12,27,28,32,29,23,34,13)(18,30,36,38,26,39,40,19);(9,12,15,29)(10,18)(11,23,25,27)(13,31,32,24)(14,28,16,34)(17,30)(19,37)(20,36)(21,26)(22,38)(33,40)(35,39);(1,3,4,7)(2,6,5,8)(9,13,14,27,15,32,16,23)(10,19,20,30,21,38,33,39)(11,12,24,28,25,29,31,34)(17,18,22,36,35,26,37,40)

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,d^c=di,f^c=fi

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, 317928);