#6454404

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(9,10,13,20,14,19,15,21)(11,22,24,33,25,34,26,16)(12,18,28,38,29,17,30,36)(23,37,27,31,39,35,40,32);(9,11,14,25)(10,16,19,33)(12,27,29,40)(13,26,15,24)(17,35,18,37)(20,34,21,22)(23,30,39,28)(31,38,32,36);(2,5)(6,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,31)(18,19)(20,36)(21,38)(22,35)(24,27)(25,39)(26,40)(32,33)(34,37);(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,32)(17,19)(20,38)(21,36)(22,37)(24,27)(25,39)(26,40)(31,33)(34,35);(1,3,4,7)(2,6,5,8)(10,19)(16,33)(17,18)(20,21)(22,34)(31,32)(35,37)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=gi,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, 6361600);