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 | |
5162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(18,19)(23,25)(24,26)(30,31);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(21,29)(24,26)(27,32)(30,31);(9,11)(10,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,7)(2,3,6,8)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137919);