Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2338 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)(5,6,8,9)(7,11,12,10)(15,19)(18,20);(5,6)(7,11)(8,9)(10,12)(13,14,17,16)(15,18,19,20)(21,22,25,24)(23,27,28,26);(6,9)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,20,19,18);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1004534);