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 | |
7112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,21)(17,22)(18,23)(20,24)(25,26)(27,28);(1,2,3,4)(13,15)(14,19)(16,22)(17,21)(18,24)(20,23)(25,27)(26,28);(6,9)(10,11)(15,24)(16,26)(19,20)(21,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,22)(17,25)(18,24)(19,23)(21,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,22)(17,19)(18,26)(20,27)(23,25)(24,28);(13,17,18,27)(14,22,23,28)(15,25,24,21)(16,19,26,20)Polycyclic presentation:
2^9|b^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650665);