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 | |
7206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,10,8,11)(13,14)(15,20)(16,24)(17,19)(18,22)(21,25)(23,26)(27,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,23)(17,22)(19,20)(21,26)(24,27)(25,28);(2,3)(14,19)(15,22)(21,24)(23,28);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,23)(17,22)(18,19)(21,27)(24,26)(25,28);(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,25)(26,28);(5,8,9,7)(6,11,12,10)(13,16,17,25)(14,21,19,24)(15,23,22,28)(18,26,20,27)Polycyclic presentation:
2^9|a^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581980);