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 | |
5104 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)(7,8)(10,13)(11,15)(17,18,22,23)(19,27,28,24)(20,25,30,32)(21,26,31,29);(1,2,5,6)(3,7,4,8)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16)(18,23)(21,31)(24,27)(25,32);(9,11,12,15)(10,14,13,16)(17,19,22,28)(18,24,23,27)(20,29,30,26)(21,25,31,32);(1,3,5,4)(2,7,6,8)(17,20)(18,25)(19,26)(21,24)(22,30)(23,32)(27,31)(28,29);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,25)(20,24)(22,31)(23,29)(27,30)(28,32)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7556349);