Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14);(1,2,5,6)(3,7,4,8)(18,22)(19,26)(20,23)(24,31)(25,29)(27,30);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,18,23,29,21,25,20,22)(19,24,30,32,27,31,26,28);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(19,27)(24,31)(26,30)(28,32);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,19,20,26,21,27,23,30)(18,24,22,28,25,31,29,32)Polycyclic presentation:
2^9|b^2=f,c^2=fhi,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313247, #72313531, #72313767 |
| maximal normal subgroup of | #72313247, #72313531, #72313767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751525);