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 | |
4967 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,24,28)(19,29,31,25)(20,22,21,26)(23,30,32,27);(1,3)(2,6)(4,8)(5,7)(10,13)(14,15)(17,19)(18,25)(20,30)(21,27)(22,23)(24,31)(26,32)(28,29);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,26,28,22)(19,30,31,27)(23,25,32,29);(1,4,5,6)(2,7,8,3)(9,11,12,16)(10,14,13,15)(17,21,24,20)(18,22,28,26)(19,30,31,27)(23,25,32,29);(1,4)(2,3)(5,6)(7,8)(17,22)(18,21)(19,23)(20,28)(24,26)(25,27)(29,30)(31,32);(1,3)(2,4)(5,7)(6,8)(17,23)(18,27)(19,22)(20,29)(21,25)(24,32)(26,31)(28,30)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8947483);