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 | |
4936 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,26,27,24)(20,21,25,31)(28,29,30,32);(1,3)(2,4)(5,8)(6,7)(10,13)(14,15)(18,23)(19,27)(20,25)(29,32);(2,5)(4,8)(9,10)(11,14)(12,13)(15,16)(19,27)(20,25)(21,31)(24,26);(1,2,6,5)(3,8,7,4)(9,11,12,16)(10,14,13,15)(17,19,22,27)(18,24,23,26)(20,30,25,28)(21,29,31,32);(1,4,6,8)(2,7,5,3)(17,20,22,25)(18,21,23,31)(19,28,27,30)(24,32,26,29);(1,5,6,2)(3,8,7,4)(17,21)(18,25)(19,29)(20,23)(22,31)(24,28)(26,30)(27,32)Polycyclic presentation:
2^9|a^2=ghi,d^2=g,e^2=gi,f^2=h,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, 8970491);