#6452957

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	#62856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3088 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)(4,7)(5,8)(9,10,14,19)(11,16,22,31)(12,17,26,34)(13,15,27,18)(20,32,23,28)(21,30,36,39)(24,33,25,29)(35,40,37,38);(1,3,5,4)(2,6,8,7)(10,15)(11,20)(12,24)(13,27)(18,19)(22,23)(25,26)(28,32)(29,33)(30,38)(35,37)(39,40);(1,3,5,4)(2,7,8,6)(9,11)(10,16)(12,21)(13,23)(14,22)(15,28)(17,30)(18,32)(19,31)(20,27)(24,35)(25,37)(26,36)(29,38)(33,40)(34,39);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,30)(18,33)(19,34)(20,35)(22,36)(23,37)(24,27)(28,38)(31,39)(32,40);(11,22)(16,31)(20,23)(21,36)(28,32)(30,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6360153);