#6575793

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

2083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=i,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, 6482989);