Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18)(19,22)(20,27,29,23)(21,24)(25,32,28,30)(26,31);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(17,19)(18,22)(20,28,29,25)(21,26)(23,30,27,32)(24,31);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25564);