Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,8,7)(9,10,14,15)(11,12,16,13)(17,18,24,29)(19,28,32,40)(20,26,35,44)(21,27,23,38)(22,37,31,25)(30,42,46,48)(33,41,36,43)(34,47,45,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,22,24,32,23,31)(18,25,38,40,29,37,27,28)(20,30,36,34,35,46,33,45)(26,39,41,48,44,47,43,42);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,33)(24,35)(25,39)(27,43)(28,42)(29,44)(31,45)(32,46)(37,47)(38,41)(40,48);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(17,21,24,23)(18,27,29,38)(19,31,32,22)(20,33,35,36)(25,40,37,28)(26,41,44,43)(30,34,46,45)(39,42,47,48);(9,13,14,12)(10,11,15,16)(17,22,21,32,24,31,23,19)(18,28,38,25,29,40,27,37)(20,34,36,46,35,45,33,30)(26,42,41,39,44,48,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ghi,d^2=gi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7515368);