Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1629 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,27,23,28,37,44)(19,29,32,40,33,43,36,24)(20,25,34,41,21,26,35,42)(30,45,47,39,31,46,48,38);(9,10)(11,15)(12,13)(14,16)(17,19,23,33)(18,24,28,40)(20,30,21,31)(22,36,37,32)(25,38,26,39)(27,43,44,29)(34,48,35,47)(41,46,42,45);(2,6)(7,8)(10,13)(14,15)(17,20,23,21)(18,25,28,26)(19,30,33,31)(22,34,37,35)(24,38,40,39)(27,41,44,42)(29,45,43,46)(32,47,36,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,26,28,25)(19,31,33,30)(22,35,37,34)(24,39,40,38)(27,42,44,41)(29,46,43,45)(32,48,36,47);(1,4,5,3)(2,8,6,7)(20,21)(25,26)(30,31)(34,35)(38,39)(41,42)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376991);