Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(6,11)(7,12)(8,9)(13,14)(15,21)(16,19)(17,20)(18,23)(22,25)(24,26)(27,28);(5,6,10,7)(8,12,9,11)(13,15)(14,18,20,21)(16,22)(17,23)(19,25,26,27)(24,28);(5,7,10,6)(8,12,9,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14651);