Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,13,14)(11,12,16,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32);(5,6,7,8)(9,11,13,16)(10,12,14,15)(18,23)(19,24)(21,30)(28,32);(5,6)(7,8)(10,14)(12,15)(17,19)(18,20)(21,31)(22,24)(23,26)(25,32)(27,30)(28,29);(1,2)(3,4)(9,12,13,15)(10,11,14,16)(17,20)(18,24)(19,23)(21,32)(22,26)(25,27)(28,30)(29,31);(9,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4040354);