#502652

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
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	#61243	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

799 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,17)(13,28)(14,32)(15,20)(16,24)(18,30)(19,22)(23,36)(25,34)(26,40)(27,35)(29,31)(33,38)(37,39);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,17)(13,29)(14,32)(15,20)(16,25)(18,22)(19,30)(23,36)(24,34)(26,40)(27,35)(28,31)(33,39)(37,38);(9,11,22,38,14,26,13,24,15,27,30,39,12,23,31,25)(10,16,29,40,17,33,18,21,20,34,28,36,32,37,19,35);(2,6)(5,7)(9,12,15,14)(10,17,20,32)(11,23,27,26)(13,22,31,30)(16,33,34,37)(18,28,19,29)(21,36,35,40)(24,38,25,39)

Polycyclic presentation:

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

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, 409848);