Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1506 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)(4,6)(5,8)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,32)(25,35)(29,43)(30,42)(31,39)(34,40)(36,47)(38,44)(41,46)(45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,30,22,32)(18,24,37,42,27,39,26,28)(20,29,34,45,36,44,35,46)(25,38,47,48,40,43,33,41);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,22,35,23,36,21,34)(18,25,26,33,27,40,37,47)(19,29,32,46,30,44,31,45)(24,38,28,41,39,43,42,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,43)(41,48)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,28,39,42)(25,33,40,47)(29,45,44,46)(38,41,43,48)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6447668);