#6454401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1904 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)(9,10,16,31,15,20,14,21)(11,22,30,17,26,38,25,34)(12,18,27,36,28,35,13,19)(23,39,24,32,29,33,40,37);(9,11)(10,17)(12,24)(13,29)(14,30)(15,26)(16,25)(18,33)(19,37)(20,34)(21,38)(22,31)(23,27)(28,40)(32,36)(35,39);(2,5)(6,8)(9,12,15,28)(10,18,20,35)(11,23,26,29)(13,16,27,14)(17,32,34,37)(19,31,36,21)(22,39,38,33)(24,25,40,30);(9,13)(10,19)(11,24)(12,16)(14,28)(15,27)(17,33)(18,31)(20,36)(21,35)(22,32)(23,25)(26,40)(29,30)(34,39)(37,38);(1,3,4,7)(2,6,5,8)(10,20)(17,34)(18,35)(19,36)(21,31)(22,38)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,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, 6361597);