Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(2,6)(7,8)(17,19,22,29)(18,23,26,32)(20,28,21,27)(24,31,25,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,32,31);(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,31,32,30);(1,4,5,3)(2,8,6,7)(18,26)(23,32)(24,25)(30,31);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535709);