Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| 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 | |
2262 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)(15,20)(16,19)(17,21)(18,22)(23,26)(24,27)(25,28);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,28)(17,24)(18,26)(19,25)(21,27)(22,23);(2,4)(13,16,18,17)(14,21,22,19)(15,23,26,20)(24,25,28,27);(5,7,8,11)(6,10,9,12)(13,17,18,16)(14,21,22,19)(15,24,26,28)(20,27,23,25);(6,9)(10,12)(14,22)(15,25)(19,21)(20,24)(23,28)(26,27)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578588);