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 |
| prime radical | #35563 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,16)(17,18,22,21)(19,23,26,24)(20,25,28,29)(27,30,32,31);(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,32)(29,31);(2,3)(6,7)(9,11,13,12)(10,15,14,16)(19,26)(20,28)(23,24)(25,29);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,16,14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30);(1,4)(2,6)(3,7)(5,8)(17,20,22,28)(18,25,21,29)(19,27,26,32)(23,30,24,31);(17,21,22,18)(19,23,26,24)(20,29,28,25)(27,30,32,31)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7651848);