Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1868 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)(5,6,9,12)(7,8,11,10)(13,14)(15,22)(16,18)(17,27)(19,21)(20,24)(23,25)(26,28);(5,7,9,11)(6,10,12,8)(13,15)(14,20)(16,22)(17,26)(18,24)(19,25)(21,28)(23,27);(2,4)(5,8,9,10)(6,11,12,7)(13,16,18,14)(15,23,24,28)(17,27,19,21)(20,25,22,26);(2,4)(6,12)(7,11)(15,24)(20,22)(23,28)(25,26);(13,17,18,19)(14,21,16,27)(15,25,24,26)(20,23,22,28)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409103);