#6454391

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

1907 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)(9,10,13,18,15,20,14,19)(11,21,23,32,25,31,24,16)(12,17,27,34,28,35,29,36)(22,33,26,39,37,40,38,30);(9,11,15,25)(10,16,20,32)(12,26,28,38)(13,24,14,23)(17,33,35,40)(18,31,19,21)(22,29,37,27)(30,36,39,34);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,26)(24,38)(25,37)(31,40)(32,39);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,26)(24,38)(25,37)(31,40)(32,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=f,b^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, 6361587);