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 | C2×#2676 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2924 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)(10,11)(12,16)(18,22)(19,25)(24,28)(27,29);(6,8)(10,11)(14,15)(17,18,21,22)(19,26,25,23)(20,28,30,24)(27,31,29,32);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,30)(29,31);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,30)(28,31);(9,12,13,16)(10,14,11,15)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)Polycyclic presentation:
2^9|b^2=f,c^2=fg,d^2=fi,e^2=g,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, 3042631);