Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 |
| prime radical | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3365 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)(11,14)(12,16)(13,15)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(6,8)(9,11,13,16)(10,14,15,12)(18,20)(19,26)(21,29)(30,31);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,23,26,25)(21,24,29,28)(27,31,32,30);(9,12)(10,11)(13,14)(15,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=f,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, 2976328);