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 | |
5571 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)(5,7)(6,8)(17,18,22,24)(19,27,29,23)(20,25,30,31)(21,32,28,26);(1,3,5,8)(2,6,7,4)(9,10,13,14)(11,15,16,12)(17,19,22,29)(18,23,24,27)(20,28,30,21)(25,32,31,26);(1,3,5,8)(2,4,7,6)(9,10)(11,15)(12,16)(13,14)(18,24)(20,30)(21,28)(23,27);(2,7)(3,8)(10,14)(11,16)(17,18)(19,27)(20,31)(21,26)(22,24)(23,29)(25,30)(28,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,32)(24,31)(26,27);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,28)(18,26,24,32)(19,20,29,30)(23,25,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,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, 8968768);