Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| 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 | |
2874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(22,25)(23,27);(2,4)(5,6,8,10)(7,9,12,11)(14,18)(15,20);(1,2,3,4)(13,14)(15,16)(17,18)(19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,24,27)(22,26,25,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|b^2=fgi,c^2=ghi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 531636);