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 | |
4843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,8,7)(10,13)(14,16)(17,18)(19,20)(21,24)(22,30)(23,25)(26,31)(27,32)(28,29);(1,3)(2,4)(5,8)(6,7)(11,15)(14,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,25)(20,24)(22,31)(27,29);(2,6)(3,8)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,19)(18,20,25,24)(22,29,31,27)(26,28,30,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840982);