#479814

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

607 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)(3,5)(4,6)(7,8)(9,10,15,29,14,19,27,18)(11,16,28,31,13,26,21,34)(12,22,24,17,25,37,39,33)(20,35,40,32,23,38,36,30);(1,3,4,7)(2,5,6,8)(9,11,15,21,14,13,27,28)(10,16,29,34,19,26,18,31)(12,23,39,36,25,20,24,40)(17,30,22,35,33,32,37,38);(9,12,15,24,14,25,27,39)(10,17,18,22,19,33,29,37)(11,20,21,36,13,23,28,40)(16,30,31,35,26,32,34,38);(2,6)(5,8)(9,13,14,11)(10,16,19,26)(12,23,25,20)(15,21,27,28)(17,32,33,30)(18,34,29,31)(22,35,37,38)(24,36,39,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^d=fi,g^d=gi

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, 387010);