Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,10)(11,12)(13,14,18,21,16,22,17,20)(15,19,25,26,23,28,24,27);(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(2,4)(5,7,9,8)(6,11,10,12)(14,20,22,21)(15,23)(19,26,28,27)(24,25);(5,8,9,7)(6,11,10,12)(14,21,22,20)(15,23)(19,27,28,26)(24,25)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=ei,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348296);