#6448748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3260 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)(11,21)(12,27)(13,18)(14,32)(15,20)(16,19)(17,25)(22,36)(23,39)(24,35)(26,34)(28,40)(29,37)(30,33)(31,38);(9,11)(10,17)(12,28)(13,30)(14,24)(15,26)(16,25)(18,36)(19,34)(20,35)(21,32)(22,29)(23,31)(27,39)(33,38)(37,40);(2,5)(6,8)(9,12,15,13)(10,18,20,27)(11,22,26,23)(14,29,16,31)(17,33,35,40)(19,37,32,38)(21,39,34,36)(24,30,25,28);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,20,27)(11,23,26,22)(14,31,16,29)(17,33,35,40)(19,37,32,38)(21,39,34,36)(24,28,25,30);(9,14,15,16)(10,19,20,32)(11,24,26,25)(12,29,13,31)(17,34,35,21)(18,37,27,38)(22,30,23,28)(33,36,40,39)

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=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, 6355944);