Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| prime radical | #30400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,31)(29,32);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,31,24,29);(10,13)(14,16)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|b^2=f,c^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72313247, #72313384, #72313579, … |
| maximal normal subgroup of | #72313247, #72313384, #72313579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7482068);