#6439253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2008 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,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, 6346449);