#410297

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(3,6)(4,5)(7,8)(9,10,15,20,16,21,14,23)(11,22,26,28,18,12,25,17)(13,29,31,35,32,37,30,19)(24,38,40,27,34,33,39,36);(2,5)(6,8)(9,11,14,25,16,18,15,26)(10,17,20,22,21,28,23,12)(13,24,30,39,32,34,31,40)(19,33,29,36,35,38,37,27);(9,12,15,28,16,22,14,17)(10,18,20,26,21,11,23,25)(13,27,31,38,32,36,30,33)(19,34,29,40,35,24,37,39);(1,3,4,7)(2,6,5,8)(9,13,14,30,16,32,15,31)(10,19,20,29,21,35,23,37)(11,24,25,39,18,34,26,40)(12,27,17,33,22,36,28,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 317493);