Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18,23,26)(19,28,27,22)(20,24,31,29)(21,25,32,30);(9,10)(11,14)(12,13)(15,16)(17,19,22,18,23,27,28,26)(20,30,21,29,31,25,32,24);(2,5)(6,8)(10,13)(14,16)(17,20)(18,24)(19,25)(21,22)(23,31)(26,29)(27,30)(28,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21)(18,25)(19,29)(20,28)(22,31)(23,32)(24,27)(26,30);(9,11,12,15)(10,14,13,16)(18,26)(19,27)(20,31)(21,32)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402796);