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 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31);(1,2,5,3)(4,6,8,7)(10,13)(14,15)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,24)(23,28)(25,30)(27,29)(31,32);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8091022);