#6454393

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	#24402	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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)(9,10,14,20,16,22,15,21)(11,23,26,35,28,34,27,17)(12,18,29,36,30,37,13,19)(24,33,25,38,31,40,39,32);(9,11)(10,17)(12,25)(13,31)(14,27)(15,26)(16,28)(18,33)(19,38)(20,34)(21,23)(22,35)(24,29)(30,39)(32,36)(37,40);(2,5)(6,8)(9,12,16,30)(10,18,22,37)(11,24,28,31)(13,14,29,15)(17,32,35,38)(19,20,36,21)(23,33,34,40)(25,27,39,26);(9,13)(10,19)(11,25)(12,14)(15,30)(16,29)(17,33)(18,20)(21,37)(22,36)(23,38)(24,27)(26,31)(28,39)(32,34)(35,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=g,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, 6361589);