Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,21)(16,25)(17,22)(18,28)(19,20)(23,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,19,22,21)(16,23,27,28)(18,24,26,25);(5,7,8,11)(6,10,9,12)(13,16,26,21,17,27,18,19)(14,20,28,24,22,15,23,25);(6,9)(10,12)(13,15)(14,21)(16,23)(17,20)(18,24)(19,22)(25,26)(27,28);(15,20)(16,27)(19,21)(24,25)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fg,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287445);