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 | |
5107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,29,31,28);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,32,29);(2,5)(6,7)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7659613);