#6576097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#577	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6483293);