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 | |
7468 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)(7,8)(9,10)(11,12)(13,14)(15,20)(16,22)(17,27)(18,28)(19,21)(23,26)(24,25);(1,2,3,4)(13,15)(14,20)(16,19)(17,23)(18,24)(21,22)(25,28)(26,27);(6,10)(7,11)(14,21)(15,16)(17,28)(24,26);(5,7,9,11)(6,8,10,12)(13,16,19,15)(14,22,21,20)(17,25,28,23)(18,26,27,24);(5,8)(6,11)(7,10)(9,12)(13,17)(14,18)(15,23)(16,25)(19,28)(20,24)(21,27)(22,26);(13,18)(14,17)(15,24)(16,26)(19,27)(20,23)(21,28)(22,25)Polycyclic presentation:
2^9|b^2=g,d^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620132);