Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3002 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)(6,8)(10,11)(13,14)(15,19)(16,18)(17,20)(21,22,25,23)(24,27,28,26);(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,20,19);(1,2,3,4)(13,14)(15,17)(16,18)(19,20)(21,23,25,22)(24,27,28,26);(5,7,9,12)(6,10,8,11)(13,14)(15,17)(16,18)(19,20)(21,24,25,28)(22,26,23,27);(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fhi,e^2=f,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, 537513);