Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,16)(15,20,26,27)(17,21,18,22)(23,25,28,24);(5,6,8,9)(7,10,11,12)(13,15,19,26)(14,20,16,27)(17,24,18,25)(21,23,22,28);(2,4)(13,16,19,14)(15,23,26,28)(17,21,18,22)(20,25,27,24);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,22)(15,24,26,25)(20,23,27,28);(6,9)(10,12)(15,25)(17,18)(20,28)(21,22)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578950);