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 | |
4859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,6,7)(10,13)(14,16)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,29,32,31);(1,3)(2,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16);(1,4)(2,6)(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(19,26)(25,28)(27,32);(1,3)(2,7)(4,8)(5,6)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(1,3)(2,4)(5,6)(7,8)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,4)(2,6)(3,8)(5,7)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876168);