#6454824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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	#62861	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2933 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)(11,22,13,19)(12,18,27,36)(14,30,15,20)(17,29,33,24)(23,34,26,35)(25,40,39,32)(28,38,31,37);(9,11,15,29,16,13,14,24)(10,17,30,19,21,33,20,22)(12,25,31,26,27,39,28,23)(18,34,38,32,36,35,37,40);(2,5)(6,8)(9,12,16,27)(10,18,21,36)(11,23,13,26)(14,28,15,31)(17,32,33,40)(19,35,22,34)(20,37,30,38)(24,39,29,25);(9,13)(10,19)(11,16)(12,26)(14,29)(15,24)(17,20)(18,35)(21,22)(23,27)(25,28)(30,33)(31,39)(32,37)(34,36)(38,40);(1,3,4,7)(2,6,5,8)(12,27)(18,36)(23,26)(25,39)(28,31)(32,40)(34,35)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6362020);