#6629196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
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	#60371	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4123 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)(4,5)(7,8)(9,10,16,22)(11,17,25,36)(12,26,29,40)(13,30,32,38)(14,20,15,21)(18,27,37,28)(19,33,39,31)(23,34,24,35);(2,5)(6,8)(9,11)(10,17)(12,13)(14,24)(15,23)(16,25)(18,19)(20,35)(21,34)(22,36)(26,30)(27,33)(28,31)(29,32)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,13)(14,27)(15,28)(16,29)(17,19)(20,26)(21,40)(22,37)(23,31)(24,33)(25,32)(30,35)(34,38)(36,39);(9,13)(10,19)(11,12)(14,31)(15,33)(16,32)(17,18)(20,38)(21,30)(22,39)(23,27)(24,28)(25,29)(26,34)(35,40)(36,37);(9,14,16,15)(10,20,22,21)(11,23,25,24)(12,27,29,28)(13,31,32,33)(17,34,36,35)(18,26,37,40)(19,38,39,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536392);