#6576749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	#60373	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2666 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,31)(24,32)(25,33)(26,35)(27,38)(28,37)(29,34)(30,36)(39,40);(1,3,5,4)(2,6,7,8)(9,11,16,25)(10,17,22,33)(12,14,24,15)(13,26,27,39)(18,32,21,19)(20,35,38,40)(23,30,28,29)(31,36,37,34);(10,18)(11,12)(13,27)(14,15)(19,32)(20,36)(21,22)(23,39)(24,25)(26,28)(31,37)(34,38);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,14)(13,28)(15,25)(16,24)(17,18)(20,37)(21,33)(22,32)(23,27)(26,29)(30,39)(31,38)(34,35)(36,40);(9,13,16,27)(10,20,22,38)(11,23,25,28)(12,26,24,39)(14,29,15,30)(17,31,33,37)(18,34,21,36)(19,35,32,40)

Polycyclic presentation:

2^9|b^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6483945);