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 | |
4918 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)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32);(1,2)(3,7)(4,8)(5,6)(17,19,23,24)(18,20,26,21)(22,30,32,27)(25,29,31,28);(10,13)(14,15)(17,20,23,21)(18,24,26,19)(22,28,32,29)(25,30,31,27);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,20)(18,24,26,19)(22,29,32,28)(25,30,31,27);(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,31)(23,26)(25,32)(27,29)(28,30);(1,4,5,3)(2,8,6,7)(17,22)(18,25)(19,27)(20,28)(21,29)(23,32)(24,30)(26,31)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=g,f^2=gi,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, 7893162);