#6439743

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
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

2071 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6346939);