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 | |
4970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,8)(4,6)(17,18)(19,24)(20,28)(21,25)(22,26)(23,29)(27,31)(30,32);(1,2)(3,4)(5,7)(6,8)(9,10,13,16)(11,12,15,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,26)(28,30)(29,32);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,16)(13,15)(20,29)(23,28)(27,32)(30,31);(2,7)(3,8)(9,12,13,14)(10,15,16,11)(17,20)(18,23)(19,27)(21,31)(22,29)(24,30)(25,32)(26,28);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,24)(20,27)(21,26)(22,25)(23,30)(28,31)(29,32);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,21,22,24)(18,25,26,19)(20,30,29,31)(23,27,28,32)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8624026);