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) = 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 | |
3008 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)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31);(6,8)(11,15)(14,16)(18,22)(19,26)(20,28)(31,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31);(1,2,3,4)(9,11,12,15)(10,14,13,16)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31);(9,10)(11,14)(12,13)(15,16)(17,20,21,28)(18,24,22,30)(19,27,26,29)(23,31,25,32)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=f,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, 3042290);