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 | |
4707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,8)(10,13)(14,16)(17,18,23,27)(19,28,22,24)(20,30,31,25)(21,26,32,29);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,28)(20,26,31,29)(21,25,32,30);(2,6)(3,8)(17,20,23,31)(18,25,27,30)(19,29,22,26)(21,28,32,24);(3,8)(4,7)(17,21)(18,26)(19,25)(20,28)(22,30)(23,32)(24,31)(27,29);(1,3)(2,7)(4,6)(5,8)(17,20)(18,25)(19,26)(21,28)(22,29)(23,31)(24,32)(27,30);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,19)(18,24,27,28)(20,29,31,26)(21,25,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,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, 7980305);