Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #9914 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9,10,11)(13,14)(15,21,23,18)(16,24,25,19)(17,20)(22,26)(27,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(2,4)(7,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28);(5,6)(7,11)(8,10)(9,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 179864);