#6448671

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	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3183 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)(9,10,16,21)(11,17,23,34)(12,18,26,37)(13,19,28,39)(14,20,15,29)(22,32,30,33)(24,38,31,35)(25,36,27,40);(9,11)(10,17)(12,24)(13,27)(14,22)(15,30)(16,23)(18,35)(19,36)(20,32)(21,34)(25,28)(26,31)(29,33)(37,38)(39,40);(2,5)(6,8)(9,12)(10,18)(11,13)(14,27)(15,25)(16,26)(17,19)(20,40)(21,37)(22,31)(23,28)(24,30)(29,36)(32,35)(33,38)(34,39);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,12)(14,24)(15,31)(16,28)(17,18)(20,38)(21,39)(22,25)(23,26)(27,30)(29,35)(32,36)(33,40)(34,37);(9,14,16,15)(10,20,21,29)(11,22,23,30)(12,25,26,27)(13,24,28,31)(17,32,34,33)(18,36,37,40)(19,38,39,35)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=i,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, 6355867);