Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #269019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11,12,9)(8,10)(13,14,17,21,18,19,16,25)(15,22,23,27,24,28,26,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(16,17)(20,27)(23,26);(2,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,24)(14,20,19,27)(16,23,17,26)(21,28,25,22);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,23)(17,18)(19,25)(20,28)(22,27)(24,26)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 176223);