#6576432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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	#24460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 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)(9,10,14,21)(11,19,25,17)(12,27,29,39)(13,20,28,34)(15,32,16,22)(18,24,38,30)(23,37,31,40)(26,35,33,36);(2,5)(6,8)(9,11)(10,17)(12,13)(14,25)(15,26)(16,33)(18,37)(19,21)(20,39)(22,35)(23,24)(27,34)(28,29)(30,31)(32,36)(38,40);(9,12,14,29)(10,18,21,38)(11,23,25,31)(13,26,28,33)(15,24,16,30)(17,34,19,20)(22,39,32,27)(35,37,36,40);(1,3,4,7)(2,6,5,8)(9,11)(10,19)(12,28)(13,29)(14,25)(15,26)(16,33)(17,21)(18,37)(20,39)(22,36)(23,30)(24,31)(27,34)(32,35)(38,40);(9,13)(10,20)(11,24)(12,26)(14,28)(15,31)(16,23)(17,18)(19,38)(21,34)(22,37)(25,30)(27,35)(29,33)(32,40)(36,39)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483628);