Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2439 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,18)(17,19)(22,25)(23,27);(5,7,9,10)(6,8,11,12)(13,15,17,16)(14,18,19,20);(1,2)(3,4)(13,14)(15,20)(16,18)(17,19)(21,22,24,25)(23,26,27,28);(5,6)(7,8)(9,11)(10,12)(13,16,17,15)(14,18,19,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1194383);