#464254

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1735 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)(3,4)(5,6)(7,8)(9,10)(11,20)(12,25)(13,27)(14,30)(15,19)(16,23)(17,26)(18,31)(21,37)(22,34)(24,38)(28,32)(29,35)(33,40)(36,39);(9,11)(10,16,19,33)(12,21)(13,28)(14,23)(15,24)(17,34,25,32)(18,20,30,38)(22,29)(26,39)(27,36,35,37)(31,40);(2,4)(9,12,15,26)(10,17,19,25)(11,21,24,39)(13,14,29,31)(16,32,33,34)(18,35,30,27)(20,36,38,37)(22,40,28,23);(6,8)(9,13,14,26,15,29,31,12)(10,17,18,27,19,25,30,35)(11,22,23,21,24,28,40,39)(16,32,20,37,33,34,38,36)

Polycyclic presentation:

2^9|b^2=g,c^2=i,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,d^b=di,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371450);