#6575780

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

2100 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,32)(24,33)(25,34)(26,36)(27,37)(28,38)(29,35)(30,40)(31,39);(2,5)(6,8)(9,11)(10,17)(12,15)(13,27)(14,24)(16,25)(18,33)(19,21)(20,37)(22,34)(23,31)(26,30)(28,29)(32,39)(35,38)(36,40);(10,18)(11,12)(14,15)(19,33)(20,35)(21,22)(23,26)(24,25)(27,28)(29,31)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12,16,24)(10,19,22,33)(11,15,25,14)(13,28,30,23)(17,21,34,18)(20,38,40,32)(26,31,27,29)(35,36,39,37);(9,13)(10,20)(11,23)(12,26)(14,29)(15,31)(16,30)(17,32)(18,35)(19,36)(21,39)(22,40)(24,27)(25,28)(33,37)(34,38)

Polycyclic presentation:

2^9|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, 6482976);