Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle, prime radical | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,13)(5,8)(6,9)(7,11)(12,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,39,37,38,42,35,40)(36,44,47,48,45,41,43,46);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,10)(13,15)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35,38,39)(34,40,42,37)(36,43,45,47)(41,48,44,46);(3,11)(7,10)(12,14)(15,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,36,38,45)(34,41,42,44)(35,43,39,47)(37,46,40,48);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,15,10,16)(18,24)(22,31)(23,27)(30,32)(36,45)(41,44)(43,47)(46,48);(1,5,6,4)(2,8,9,13)(3,12,11,14)(7,16,10,15)(33,37,39,42,38,40,35,34)(36,46,47,44,45,48,43,41)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=cfgi,d^b=df,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315277);