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 | |
4755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28);(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,16,15)(19,26)(20,23)(27,29)(28,31);(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,26)(24,28)(27,30)(31,32);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,32,29);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,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, 8094096);