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 | |
7404 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)(7,11)(10,12)(13,14)(15,21)(16,24)(17,22)(18,19)(20,25)(23,26)(27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,23)(18,22)(19,21)(20,26)(24,27)(25,28);(6,9)(10,12)(14,19)(15,22)(16,25)(26,27);(13,14)(15,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,25)(14,20,19,24)(15,23,22,28)(17,26,21,27);(5,7)(6,10)(8,11)(9,12)(13,17)(14,15)(16,26)(18,21)(19,22)(20,23)(24,28)(25,27)Polycyclic presentation:
2^9|b^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615817);