Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1948 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)(15,22)(19,27)(23,28)(24,26);(5,7,9,11)(6,10,12,8)(13,14)(15,23)(16,20)(17,21)(18,25)(19,24)(22,28)(26,27);(2,4)(5,8,9,10)(6,11,12,7)(13,15)(14,19)(16,24)(17,22)(18,26)(20,28)(21,27)(23,25);(2,4)(6,12)(7,11);(13,16,17,18)(14,20,21,25)(15,24,22,26)(19,28,27,23)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409083);