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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2250 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)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28);(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,26)(25,32)(27,29);(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27);(9,11)(10,13)(12,16)(14,15)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=g,d^2=f,e^2=f,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, 4040212);