Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3078 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)(19,24)(20,29)(21,25)(22,23)(26,27)(28,31)(30,32);(5,6,7,8)(9,11,14,13)(10,15,12,16)(18,23)(20,25)(24,31)(26,32);(5,6)(7,8)(9,10,14,12)(11,16,13,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,32)(27,29);(1,2,3,4)(9,12,14,10)(11,15,13,16)(17,20)(18,21)(19,26)(22,25)(23,29)(24,30)(27,31)(28,32);(9,13,14,11)(10,15,12,16)(17,21)(18,25)(19,27)(20,23)(22,29)(24,26)(28,30)(31,32)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3372385);