Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(14,18)(15,22)(16,19)(20,25)(23,24)(26,28);(5,7)(6,9)(8,11)(10,12)(13,14,19,25,17,20,16,18)(15,21,22,28,24,27,23,26);(2,4)(7,11)(9,12)(14,20)(18,25)(21,27)(26,28);(13,15)(14,21)(16,23)(17,24)(18,26)(19,22)(20,27)(25,28)Polycyclic presentation:
2^9|a^2=gi,b^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308819);