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 | |
5116 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)(5,6)(7,8)(10,13)(14,16)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30);(1,3)(2,4)(5,7)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32);(3,7)(4,8)(17,19,22,27)(18,23,25,20)(21,29,31,26)(24,28,32,30);(2,6)(4,8)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31);(1,4)(2,3)(5,8)(6,7)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7971251);