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 | |
4945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,4,7)(10,13)(14,15)(17,18,22,20)(19,24,26,23)(21,28,29,25)(27,30,32,31);(1,2)(3,6)(4,7)(5,8)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32);(9,10)(11,15)(12,13)(14,16)(18,20)(19,26)(25,28)(27,32);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32);(1,4,5,3)(2,7,8,6)(17,20,22,18)(19,23,26,24)(21,28,29,25)(27,30,32,31);(17,21,22,29)(18,25,20,28)(19,27,26,32)(23,30,24,31)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586891);