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 | |
4700 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)(3,7)(10,13)(14,16)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31);(3,7)(4,8)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,3)(2,4)(5,7)(6,8)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,4)(2,3)(5,8)(6,7)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=h,e^2=h,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, 7982637);