#6448664

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
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3194 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)(9,10,15,20)(11,16,23,31)(12,17,26,34)(13,19,14,18)(21,29,38,40)(22,35,27,30)(24,39,37,32)(25,36,28,33);(9,11)(10,16)(12,24)(13,22)(14,27)(15,23)(17,32)(18,30)(19,35)(20,31)(21,25)(26,37)(28,38)(29,33)(34,39)(36,40);(2,5)(6,8)(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,33)(19,36)(20,34)(22,37)(23,38)(24,27)(30,39)(31,40)(32,35);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,33)(19,36)(20,34)(22,37)(23,38)(24,27)(30,39)(31,40)(32,35);(9,13,15,14)(10,18,20,19)(11,22,23,27)(12,25,26,28)(16,30,31,35)(17,33,34,36)(21,37,38,24)(29,39,40,32)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 6355860);