#430917

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fi,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338113);