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 | |
5066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,8)(6,7)(10,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,28)(29,31)(30,32);(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,27)(21,26)(22,29)(23,28)(25,32)(30,31);(2,7)(3,8)(18,20)(19,28)(25,29)(26,30);(1,3,6,8)(2,4,7,5)(17,20,23,18)(19,27,28,24)(21,29,31,25)(22,30,32,26);(1,4,6,5)(2,3,7,8)(9,11,13,12)(10,15,14,16)(17,21)(18,25)(19,26)(20,29)(22,27)(23,31)(24,32)(28,30);(1,5,6,4)(2,8,7,3)(9,12,13,11)(10,16,14,15)(17,22)(18,26)(19,29)(20,30)(21,24)(23,32)(25,28)(27,31)Polycyclic presentation:
2^9|d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7602117);