#6448669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3156 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)(9,10,15,11)(12,16,24,19)(13,17,27,20)(14,21,31,18)(22,29,33,25)(23,32,37,35)(26,30,34,28)(36,38,40,39);(9,11,15,10)(12,22,24,33)(13,26,27,34)(14,18,31,21)(16,25,19,29)(17,28,20,30)(23,35,37,32)(36,39,40,38);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,29)(15,24)(17,32)(18,22)(20,35)(21,33)(25,31)(26,36)(27,37)(28,39)(30,38)(34,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,20)(12,23)(14,30)(15,27)(16,32)(18,26)(19,35)(21,34)(22,36)(24,37)(25,39)(28,31)(29,38)(33,40);(12,24)(16,19)(22,33)(23,37)(25,29)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6355865);