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 | |
4691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,6,7,8)(10,13)(14,16)(17,18,22,23)(19,25,27,20)(21,24,29,31)(26,32,30,28);(9,10,12,13)(11,14,15,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32);(2,4)(6,8)(18,23)(20,25)(21,29)(26,30);(3,7)(6,8)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,29)(28,31);(1,3)(2,6)(4,8)(5,7)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=h,b^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,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, 7973747);