Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1532 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)(11,14)(12,13)(15,16)(18,23)(24,31)(25,26)(27,32);(2,5)(6,8)(17,18,21,23)(19,27,28,32)(20,26,22,25)(24,29,31,30);(10,13)(14,16)(17,19,21,28)(18,24,23,31)(20,30,22,29)(25,27,26,32);(1,3,4,7)(2,6,5,8)(18,25)(19,28)(20,22)(23,26)(24,32)(27,31);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,26,23,25)(19,29,28,30)(24,27,31,32)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363204);