#6439620

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	#60426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2285 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)(11,20)(12,17)(13,28)(14,19)(15,18)(16,22)(21,35)(23,31)(24,34)(25,26)(27,32)(29,33)(30,36)(37,40)(38,39);(2,5)(6,8)(9,10)(11,20)(12,25)(13,19)(14,28)(15,18)(16,23)(17,26)(21,30)(22,31)(24,34)(27,33)(29,32)(35,36)(37,39)(38,40);(9,11,15,24)(10,16,18,31)(12,21,27,37)(13,23,14,22)(17,30,32,39)(19,34,28,20)(25,35,33,40)(26,38,29,36);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,32)(19,33)(20,35)(22,36)(23,38)(24,37)(25,28)(31,39)(34,40);(10,18)(16,31)(17,32)(19,28)(20,34)(25,33)(30,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^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,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, 6346816);