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 | |
4927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,16,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32);(1,3,5,8)(2,6,4,7)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31);(9,10)(11,15)(12,14)(13,16)(18,23)(19,24)(21,30)(28,32);(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32);(1,2,5,4)(3,6,8,7)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32);(1,4,5,2)(3,7,8,6)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590493);