#6449090

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356286);