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 | #45244 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,27)(21,26)(23,28)(24,31)(25,30)(29,32);(10,14)(11,16)(17,19,22,18)(20,25,29,23)(21,26,31,24)(27,28,32,30);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,29)(18,23,19,25)(21,30,31,28)(24,32,26,27);(2,5)(6,8)(10,14)(13,15)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^9|c^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7455801);