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 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18,22,20)(19,25,26,23)(21,24,30,28)(27,32,29,31);(1,2,5,6)(3,7,4,8)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31);(9,10)(11,15)(12,14)(13,16)(19,26)(23,25)(27,29)(31,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,25,26,23)(21,24,30,28)(27,31,29,32);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32);(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589688);