Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1442 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,14)(10,15,13,16)(19,25,27,28)(22,30,26,31);(1,3)(2,6)(4,13,14,10)(5,11)(7,16,12,15)(8,9)(17,18,20,23,21,24,29,32)(19,26,25,31,27,22,28,30);(1,4)(2,7,8,12)(3,10)(5,14)(6,15,9,16)(11,13)(17,19)(18,22)(20,28)(21,27)(23,31)(24,26)(25,29)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,e^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,e^c=ei,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28226);