Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6502937 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(9,10,14,20,16,21,15,36)(11,22,25,40,26,39,35,17)(12,27,29,42,31,54,30,18)(13,19,33,44,32,43,34,60)(23,37,48,45,49,62,58,52)(24,46,51,63,50,64,59,38)(28,53,56,65,55,70,57,41)(47,61,68,66,67,72,71,69);(1,2)(3,6)(4,5)(7,8)(9,11,16,26)(10,17,21,40)(12,23,31,49)(13,24,32,50)(14,35,15,25)(18,37,42,62)(19,38,43,63)(20,39,36,22)(27,52,54,45)(28,47,55,67)(29,58,30,48)(33,59,34,51)(41,61,65,72)(44,64,60,46)(53,69,70,66)(56,71,57,68);(2,5)(3,7)(9,12,14,29,16,31,15,30)(10,18,36,54,21,42,20,27)(11,23,35,58,26,49,25,48)(13,28,33,56,32,55,34,57)(17,37,22,45,40,62,39,52)(19,41,60,70,43,65,44,53)(24,47,59,71,50,67,51,68)(38,61,46,66,63,72,64,69);(1,3,4,7)(2,6,5,8)(9,13,16,32)(10,19,21,43)(11,24,26,50)(12,28,31,55)(14,33,15,34)(17,38,40,63)(18,41,42,65)(20,44,36,60)(22,46,39,64)(23,47,49,67)(25,51,35,59)(27,53,54,70)(29,56,30,57)(37,61,62,72)(45,66,52,69)(48,68,58,71);(2,5)(6,8)(13,32)(19,43)(24,50)(28,55)(33,34)(38,63)(41,65)(44,60)(46,64)(47,67)(51,59)(53,70)(56,57)(61,72)(66,69)(68,71)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427241);