#6438610

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

2156 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)(11,16)(12,25)(13,18)(14,19)(15,21)(17,28)(20,32)(22,34)(23,36)(24,35)(26,39)(27,38)(29,37)(30,33)(31,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,28)(14,23)(15,24)(17,18)(19,36)(20,34)(21,35)(22,32)(25,39)(27,31)(29,30)(33,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,13)(14,27)(15,29)(16,33)(18,36)(19,25)(20,37)(21,38)(22,30)(23,26)(24,31)(28,32)(34,40)(35,39);(9,13)(10,18)(11,12)(14,30)(15,31)(16,25)(17,36)(19,33)(20,39)(21,40)(22,27)(23,28)(24,29)(26,32)(34,38)(35,37);(9,14,15,32)(10,19,21,20)(11,22,24,23)(12,27,29,28)(13,30,31,26)(16,34,35,36)(17,25,38,37)(18,33,40,39)

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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, 6345806);