#6435583

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,g^c=gi

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, 6342779);