#6454397

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361593);