Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(10,13)(11,15)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28);(6,8)(17,19,22,24)(18,23,26,20)(21,30,31,27)(25,28,29,32);(1,2,4,3)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,32)(28,29)(30,31);(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1052832);