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 | |
4901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(10,13)(14,16)(17,18)(19,21)(20,28)(22,24)(23,26)(25,30)(27,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,23,24,21)(20,29,30,27)(25,31,28,32);(2,3)(6,7)(18,24)(19,26)(20,30)(31,32);(17,18)(19,23)(20,25)(21,26)(22,24)(27,32)(28,30)(29,31);(1,3,5,2)(4,7,8,6)(17,20,22,30)(18,25,24,28)(19,27,26,29)(21,31,23,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,24,19)(20,31,30,32)(25,29,28,27)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625293);