#6452036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2564 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)(3,8)(4,6)(5,7)(9,10,15,21,16,22,14,20)(11,23,26,29,27,40,37,17)(12,25,30,38,31,18,32,33)(13,19,35,28,36,24,34,39);(9,11,16,27)(10,17,22,29)(12,28,31,39)(13,33,36,38)(14,26,15,37)(18,34,25,35)(19,32,24,30)(20,23,21,40);(1,3,5,4)(2,6,7,8)(9,12,16,31)(10,18,22,25)(11,24,27,19)(13,29,36,17)(14,32,15,30)(20,38,21,33)(23,35,40,34)(26,39,37,28);(1,4,5,3)(2,6,7,8)(9,13,16,36)(10,19,22,24)(11,25,27,18)(12,29,31,17)(14,34,15,35)(20,39,21,28)(23,30,40,32)(26,38,37,33);(9,14,16,15)(10,20,22,21)(11,26,27,37)(12,30,31,32)(13,34,36,35)(17,23,29,40)(18,33,25,38)(19,39,24,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6359232);