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 | |
5030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,16)(17,18,23,25)(19,24,27,32)(20,28,30,26)(21,31,29,22);(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,32)(26,31);(2,6)(3,7)(18,25)(19,27)(20,30)(22,31);(1,3)(2,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,21)(22,24)(23,30)(25,28)(27,29)(31,32);(1,4)(2,3)(5,8)(6,7)(17,21)(18,22)(19,20)(23,29)(24,26)(25,31)(27,30)(28,32);(17,22)(18,21)(19,26)(20,24)(23,31)(25,29)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713839);