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 | |
4916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,7)(10,13)(14,16)(17,18,22,26)(19,25,27,23)(20,21,24,31)(28,29,32,30);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,26)(29,31);(1,2)(3,4)(5,6)(7,8)(19,27)(23,25)(28,32)(29,30);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,20)(18,21)(19,28)(22,24)(23,30)(25,29)(26,31)(27,32);(1,3,5,7)(2,4,6,8)(17,21)(18,24)(19,29)(20,26)(22,31)(23,28)(25,32)(27,30);(1,4)(2,7)(3,6)(5,8)(17,19)(18,25)(20,28)(21,29)(22,27)(23,26)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7699415);