#6483291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473165
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1922 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)(11,21)(12,25)(13,20)(14,19)(15,18)(16,24)(17,29)(22,33)(23,36)(26,31)(27,35)(28,34)(30,32)(37,38)(39,40);(3,4)(6,8)(9,11)(10,16)(12,26)(13,30)(14,24)(15,23)(17,33)(18,32)(19,36)(20,21)(22,27)(25,39)(28,38)(29,40)(31,34)(35,37);(1,3,5,4)(2,6,7,8)(9,12,13,28,15,27,14,29)(10,17,19,35,18,34,20,25)(11,22,30,40,23,26,24,38)(16,31,36,39,32,33,21,37);(1,4,5,3)(2,6,7,8)(11,23)(12,27)(16,32)(17,34)(21,36)(24,30)(25,35)(28,29);(2,7)(6,8)(10,18)(12,27)(16,32)(19,20)(21,36)(22,26)(28,29)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390487);