#6448667

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	#23400	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3226 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)(9,10,14,18)(11,15,21,30)(12,16,23,32)(13,24,25,17)(19,33,26,28)(20,29,36,39)(22,34,27,31)(35,40,37,38);(11,19)(12,22)(13,25)(15,28)(16,31)(17,24)(21,26)(23,27)(30,33)(32,34)(35,37)(38,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,33)(18,30)(19,25)(22,35)(23,36)(24,28)(27,37)(31,38)(32,39)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,27)(14,23)(15,29)(17,34)(18,32)(19,35)(21,36)(22,25)(24,31)(26,37)(28,38)(30,39)(33,40);(11,21)(15,30)(19,26)(20,36)(28,33)(29,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^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, 6355863);