#501453

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(3,8)(4,6)(5,7)(9,10,17,28,13,21,31,33,14,12,15,20,16,19,29,30)(11,22,32,34,24,23,37,40,25,18,26,36,27,35,39,38);(1,3,5,4)(2,6,7,8)(9,11,14,25)(10,18,12,22)(13,24,16,27)(15,26,17,32)(19,23,21,35)(20,34,28,36)(29,39,31,37)(30,40,33,38);(1,4,5,3)(2,6,7,8)(9,12,14,10)(11,23,25,35)(13,21,16,19)(15,30,17,33)(18,24,22,27)(20,31,28,29)(26,36,32,34)(37,38,39,40);(2,7)(6,8)(9,13,14,16)(10,19,12,21)(11,24,25,27)(15,29,17,31)(18,23,22,35)(20,33,28,30)(26,39,32,37)(34,38,36,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,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, 408649);