#479808

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

598 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=ef,d^2=e,f^2=i,g^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=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387004);