Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14)(15,20)(16,24)(17,21)(18,23)(19,25)(22,26)(27,28);(1,2)(3,4)(5,6,10,9)(7,12,8,11);(5,7,10,8)(6,11,9,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(13,16,17,25)(14,19,21,24)(15,22,23,28)(18,26,20,27);(5,8,10,7)(6,11,9,12)(13,15)(14,20)(16,22)(17,23)(18,21)(19,27)(24,26)(25,28);(5,9,10,6)(7,12,8,11)(13,14)(15,18)(16,19)(17,21)(20,23)(22,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562471);