Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(6,9)(10,12)(14,17)(18,19)(21,23,24,27)(22,25,26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,18,17,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,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, 4514378);