#6450108

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2919 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^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, 6357304);