Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2660 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)(13,14,18,19)(15,21,22,26)(16,24,25,23)(17,27,28,20);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,24)(18,22)(19,25)(20,26)(21,27)(23,28);(2,4)(6,9)(10,12)(14,19)(15,22)(20,27)(23,24);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,21)(18,25)(19,22)(20,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311197);