#457241

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1374 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=ei,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,g^b=gi,d^c=de,e^c=ei,h^c=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, 364437);