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 | |
7123 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)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,23)(17,19)(20,24)(21,26)(22,28)(25,27);(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,18,19,26)(16,22,24,27)(20,25,23,28);(2,4)(14,19)(15,21)(20,23)(22,27);(1,2)(3,4)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(13,14)(15,18)(16,20)(17,19)(21,26)(22,25)(23,24)(27,28);(5,7,8,11)(6,9,10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,23)(21,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580320);