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 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(16,24)(20,27)(22,28)(25,26);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,21)(18,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,22)(17,23)(19,21)(20,25)(24,28)(26,27);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,14)(15,18)(16,20)(17,21)(19,23)(22,25)(24,27)(26,28);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537866);