Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 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)(14,18)(15,21)(16,24)(19,23)(22,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(6,9)(10,12)(13,15,18,21,17,23,14,19)(16,22,25,27,24,28,20,26);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27);(16,24)(20,25)(22,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282207);