#6439607

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	#60422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2241 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:

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

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,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, 6346803);