Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #35715 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,23)(21,22)(25,30)(26,28)(27,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,27)(18,24,21,31)(20,29,30,26)(23,32,25,28);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20)(18,25)(19,26)(21,23)(22,30)(24,32)(27,29)(28,31);(10,14)(13,16)(17,21,22,18)(19,26,27,29)(20,25,30,23)(24,32,31,28);(2,7)(6,8)(18,21)(20,30)(24,31)(26,29)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72285056 |
| maximal normal subgroup of | #72285056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7431821);