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 | |
8366 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)(6,9)(10,11)(13,14)(15,19)(16,25)(17,21)(18,22)(20,26)(23,27)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,17)(18,24)(20,27)(21,25)(22,28)(23,26);(2,4)(5,6)(7,11)(8,9)(10,12)(13,16,18,26)(14,20,22,25)(15,23,24,17)(19,21,28,27);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,17)(18,26)(19,27)(21,28)(22,25)(23,24);(13,17)(14,21)(15,16)(18,23)(19,25)(20,28)(22,27)(24,26);(13,14)(15,19)(16,20)(17,27)(18,22)(21,23)(24,28)(25,26)Polycyclic presentation:
2^9|c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603721);