Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,22,16,24)(18,28,27,21)(19,26,25,23);(1,2)(3,4)(6,9)(7,12)(13,14)(15,19)(16,25)(17,20)(18,21)(22,23)(24,26)(27,28);(2,4)(13,15,17,16)(14,19,20,25)(18,26,27,23)(21,24,28,22);(5,7,8,12)(6,10,9,11)(13,16,17,15)(14,19,20,25)(18,23,27,26)(21,24,28,22);(14,20)(19,25)(21,28)(22,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409509);