Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(9,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(16,17)(20,25)(22,27)(23,26);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,24)(14,19,21,28)(16,23,17,26)(20,27,25,22);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,23)(17,18)(19,27)(21,25)(22,28)(24,26)Polycyclic presentation:
2^9|b^2=hi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156129);