Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,19,23,18,16,20,27)(15,24,22,17,25,21,26,28);(5,7)(6,9)(8,11)(10,12)(13,15,20,22,18,25,19,26)(14,21,23,17,16,24,27,28);(7,11)(9,12)(13,16)(14,18)(15,24)(17,26)(19,23)(20,27)(21,25)(22,28);(2,4)(13,17,20,24,18,28,19,21)(14,22,23,25,16,26,27,15)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310322);