Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,20,21,23,30,28)(19,24,27,29,25,31,32,26);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(18,23)(19,25)(20,28)(27,32);(10,14)(11,15)(17,19)(18,24)(20,29)(21,25)(22,27)(23,31)(26,28)(30,32);(3,8)(5,7)(10,14)(13,16)(17,20,22,23,21,28,30,18)(19,26,27,24,25,29,32,31);(2,6)(5,7)(19,25)(24,31)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=df,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282580, #72282683 |
| maximal normal subgroup of | #72282580, #72282683 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6329263);