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 | |
4944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,4,5)(18,22)(19,26)(25,30)(27,28);(1,3)(2,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,25)(21,22)(24,26)(27,31)(28,32)(29,30);(1,3)(2,7)(4,6)(5,8)(10,13)(11,16)(17,19,21,26)(18,23,22,24)(20,28,29,27)(25,32,30,31);(2,8)(3,4)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,28)(21,26)(22,23)(25,31)(27,29)(30,32);(1,4,6,3)(2,5,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,29)(18,25,22,30)(19,27,26,28)(23,31,24,32);(1,5)(2,3)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,22)(23,26)(27,32)(28,31)(29,30)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969408);