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 | |
5146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(5,7)(10,13)(11,15)(20,27)(24,29)(26,31)(30,32);(2,4)(3,8)(9,10,12,13)(11,14,15,16)(18,22)(19,25)(24,29)(26,31);(1,2,6,4)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,6,8)(2,7,4,5)(9,10)(11,14)(12,13)(15,16);(1,4,6,2)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,5)(2,3)(4,8)(6,7)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015942);