Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3022 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)(10,13)(11,15)(17,18)(19,22)(20,25)(21,24)(23,27)(26,29)(28,32)(30,31);(6,8)(11,15)(14,16)(18,19)(20,27)(21,29)(31,32);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,31,30,32);(1,2,3,4)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,30)(22,27)(24,32)(26,31)(28,29);(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,31)(25,32)(27,30)Polycyclic presentation:
2^9|c^2=h,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3043409);