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 | |
5007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,24,29,23)(20,21,25,31)(27,28,30,32);(1,3,6,5)(2,7,8,4)(9,10,12,14)(11,13,16,15)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,28,31,32);(1,2)(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16);(1,4)(2,5)(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,30)(31,32);(1,4)(2,3)(5,8)(6,7)(17,20)(18,21)(19,27)(22,25)(23,32)(24,28)(26,31)(29,30);(1,5,6,3)(2,7,8,4)(17,21)(18,25)(19,28)(20,26)(22,31)(23,27)(24,30)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9844914);