#499204

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406400);