#6576102

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×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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)(9,10,15,21)(11,16,27,37)(12,13,18,31)(14,20,33,19)(17,32,30,29)(22,38,39,40)(23,35,34,28)(24,26,36,25);(2,5)(6,8)(9,11)(10,16)(12,28)(13,23)(14,25)(15,27)(17,38)(18,35)(19,36)(20,24)(21,37)(22,29)(26,33)(30,40)(31,34)(32,39);(9,12,15,18)(10,17,21,30)(11,22,27,39)(13,19,31,20)(14,32,33,29)(16,34,37,23)(24,38,36,40)(25,28,26,35);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,23)(12,21)(14,30)(15,31)(16,35)(17,33)(19,32)(20,29)(22,24)(25,40)(26,38)(27,34)(28,37)(36,39);(9,10)(11,24)(12,17)(13,29)(14,19)(15,21)(16,25)(18,30)(20,33)(22,38)(23,35)(26,37)(27,36)(28,34)(31,32)(39,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,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, 6483298);