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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1528 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(25,31)(29,30);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,27,26,32)(20,25,22,31)(24,29,28,30);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,18)(19,28)(20,31)(21,23)(22,25)(24,26)(27,29)(30,32);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,19)(18,24)(20,30)(21,26)(22,29)(23,28)(25,32)(27,31);(2,8)(6,7)(17,20,21,22)(18,25,23,31)(19,29,26,30)(24,27,28,32)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fi,d^2=fi,e^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433866);