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 | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,38,42,37,39,36,41)(35,43,45,48,44,40,46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(34,39)(35,44)(41,42)(45,46);(3,10)(6,9)(11,16)(13,15)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(35,44)(40,43)(45,46)(47,48);(2,7)(3,10)(8,14)(11,16)(18,24)(22,31)(23,27)(30,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,43)(41,48)(42,47);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(33,36,37,38)(34,41,39,42)(35,45,44,46)(40,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=df,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314923);