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 | |
8146 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)(6,9)(10,11)(13,14)(15,20)(16,17)(18,22)(19,25)(21,27)(23,24)(26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,22)(21,23)(27,28);(5,6)(7,11)(8,9)(10,12)(13,15,18,25)(14,19,22,20)(16,26,21,23)(17,24,27,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,20)(16,23)(17,24)(18,25)(19,22)(21,26)(27,28);(13,16,18,21)(14,17,22,27)(15,23,25,26)(19,28,20,24);(13,17)(14,21)(15,24)(16,22)(18,27)(19,23)(20,26)(25,28)Polycyclic presentation:
2^9|b^2=gh,c^2=i,e^2=i,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, 7585117);