Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18,22,26,23,20,21,25)(19,27,30,24,31,32,29,28);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19,22,30,23,31,21,29)(18,24,26,32,20,28,25,27);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,20)(18,23)(19,28)(21,26)(22,25)(24,31)(27,30)(29,32);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fi,e^c=ei,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99688);