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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1252 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,35)(22,26)(23,38)(24,30)(25,34)(27,37)(29,44)(32,43)(33,47)(36,42)(39,48)(40,46)(41,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,26,28)(20,32,33,48)(21,23,36,37)(25,40,41,44)(27,42,38,35)(29,45,46,34)(31,39,47,43);(1,3,5,4)(2,7,6,8)(17,20,23,34,22,33,37,45)(18,25,27,31,26,41,38,47)(19,29,21,32,30,46,36,48)(24,39,35,40,28,43,42,44);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,21,23,30,22,36,37,19)(18,24,27,35,26,28,38,42)(20,32,34,46,33,48,45,29)(25,39,31,40,41,43,47,44);(1,4,5,3)(2,7,6,8)(18,26)(20,33)(24,28)(27,38)(29,46)(32,48)(34,45)(35,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fgi,d^2=fgi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7501152);