Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(7,11)(13,14,16,24,19,23,18,21)(15,22,25,20,26,28,17,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,23,27)(16,17,18,25)(21,28,24,22);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,24)(15,25,26,17)(20,22,27,28);(6,9)(10,12)(13,17,18,26,19,25,16,15)(14,22,21,27,23,28,24,20)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=f,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338533);