Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(15,16)(17,18)(19,27)(20,22)(21,26)(23,25)(24,30)(28,32)(29,31);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,30)(18,24,25,27)(20,21,28,31)(22,29,32,26);(3,7)(4,8)(9,11,13,12)(10,15,14,16)(18,25)(22,32)(24,27)(26,29);(2,6)(4,8)(9,12,13,11)(10,16,14,15)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,27)(30,31);(1,4)(2,3)(5,8)(6,7)(17,21)(18,26)(19,28)(20,30)(22,24)(23,31)(25,29)(27,32);(1,2)(3,4)(5,6)(7,8)(17,22)(18,20)(19,29)(21,24)(23,32)(25,28)(26,30)(27,31)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7970349);