#6576751

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	#60375	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2580 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,28)(13,19)(14,20)(15,22)(16,23)(18,30)(21,24)(25,35)(26,36)(27,37)(29,39)(31,38)(32,34)(33,40);(1,3,5,4)(2,6,7,8)(9,11,16,27)(10,17,23,37)(12,29,25,32)(13,24,26,15)(14,31,33,30)(18,20,38,40)(19,21,36,22)(28,39,35,34);(9,12,16,25)(10,18,23,38)(11,14,27,33)(13,29,26,32)(15,31,24,30)(17,34,37,39)(19,20,36,40)(21,35,22,28);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,24)(12,14)(15,27)(16,26)(17,21)(18,34)(20,28)(22,37)(23,36)(25,33)(29,31)(30,32)(35,40)(38,39);(9,14,16,33)(10,20,23,40)(11,25,27,12)(13,31,26,30)(15,32,24,29)(17,35,37,28)(18,19,38,36)(21,39,22,34)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,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, 6483947);