Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18,20,24,22,26,21,25)(19,27,29,32,28,31,30,23);(2,4)(5,6)(7,8)(10,13)(14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(2,4)(6,8)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(9,11,12,15)(10,14,13,16)(19,28)(23,32)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239566, #72239613, #72239634, … |
| maximal normal subgroup of | #72239566, #72239613, #72239634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265859);