#6445808

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
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	C₂⁴	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6353004);