Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(14,17)(15,19)(21,22,26,27)(23,28,25,24);(6,10)(7,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,25)(22,24,27,28);(6,10)(8,11)(15,19)(18,20)(22,27)(23,25);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,27)(26,28);(5,8,9,11)(6,12,10,7)(13,15,16,19)(14,18,17,20)(21,25,26,23)(22,24,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508156);