#465209

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1076 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)(6,8)(9,10)(11,23)(12,26)(13,29)(14,20)(15,22)(16,21)(17,18)(19,32)(24,27)(25,28)(30,35)(31,37)(33,36)(34,40)(38,39);(1,2,4,5)(3,6,7,8)(9,11,16,26,14,18,15,25)(10,17,21,28,20,23,22,12)(13,24,33,40,31,35,32,39)(19,34,29,30,36,38,37,27);(3,7)(6,8)(9,12,15,17,14,28,16,23)(10,18,21,26,20,11,22,25)(13,30,32,38,31,27,33,34)(19,24,29,39,36,35,37,40);(1,3,4,7)(2,6,5,8)(9,13,14,31)(10,19,20,36)(11,24,18,35)(12,27,28,30)(15,32,16,33)(17,34,23,38)(21,29,22,37)(25,39,26,40)

Polycyclic presentation:

2^9|b^2=egi,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=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, 372405);