Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1978 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)(7,12)(8,11)(9,10)(13,14,16,18)(15,17,19,20)(21,22,25,23)(24,27,28,26);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(15,19)(17,20);(1,2)(3,4)(6,9)(7,11)(13,14)(15,17)(16,18)(19,20)(21,23,25,22)(24,27,28,26);(5,8,10,12)(6,11,9,7)(13,15)(14,17)(16,19)(18,20)(21,24)(22,26)(23,27)(25,28);(5,9,10,6)(7,12,11,8)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2135396);