Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #2540 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,15)(12,18)(14,20)(17,19)(21,22,25,28,24,26,31,33)(23,27,30,34,29,32,35,36);(1,2)(3,4)(7,14)(10,19)(13,20)(15,17)(22,26)(23,29)(28,33)(30,35);(5,7,9,14)(6,10,11,19)(8,17,18,15)(12,20,16,13)(21,23)(22,27)(24,29)(25,30)(26,32)(28,34)(31,35)(33,36);(2,4)(6,11)(10,19)(12,16)(13,20)(22,26)(23,29)(28,33)(30,35)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=dei,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127036);