Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,7)(17,18,22,24)(19,21,26,31)(20,27,30,25)(23,28,32,29);(1,2,4,5)(3,7,8,6)(9,10)(11,14)(12,13)(15,16);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,19,21,24,22,26,31,18)(20,25,28,32,30,27,29,23);(3,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,30)(24,32)(26,27)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377024);