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 | |
4922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,4,7)(10,13)(11,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,2)(3,6)(4,7)(5,8)(10,13)(14,16)(19,26)(23,25)(27,32)(30,31);(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,28)(30,31);(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3,5,4)(2,6,8,7)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4,5,3)(2,7,8,6)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550774);