#6454395

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	#3006	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)(9,10,13,19,14,20,15,21)(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)(10,16,20,33)(12,27,29,40)(13,26,15,24)(17,35,18,37)(19,34,21,22)(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)(18,20)(19,36)(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)(17,20)(19,38)(21,36)(22,37)(24,27)(25,39)(26,40)(31,33)(34,35);(1,3,4,7)(2,6,5,8)

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,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, 6361591);