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 | |
5304 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)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,25)(21,27)(22,24)(23,29)(28,31)(30,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,25)(22,29)(24,26)(27,30)(31,32);(3,7)(4,8)(9,11,12,15)(10,13,14,16)(18,24)(21,31)(23,26)(25,32);(2,6)(4,8)(17,20,22,30)(18,25,24,32)(19,28,29,27)(21,26,31,23);(1,4)(2,3)(5,8)(6,7)(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,30)(31,32);(1,2)(3,4)(5,6)(7,8)(17,21)(18,27)(19,25)(20,26)(22,31)(23,30)(24,28)(29,32)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=h,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, 7970308);