Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,26)(25,28);(5,7,8,12)(6,9,10,11)(15,23)(20,27)(24,25)(26,28);(7,12)(9,11)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(6,10)(9,11)(13,15)(14,20)(16,25)(17,24)(18,23)(19,26)(21,28)(22,27);(1,3,4,2)(13,16,18,17)(14,21,22,19)(15,24,23,25)(20,26,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^c=ei,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285031);