Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 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)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,28,23,29)(19,30,32,41,33,24,34,40)(20,26,37,25,36,43,35,42)(31,45,48,44,47,39,46,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,30)(20,31,36,47)(21,32,23,34)(25,38,42,44)(26,39,43,45)(27,40,29,41)(35,46,37,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,35)(22,36)(23,37)(24,38)(26,27)(28,42)(29,43)(30,44)(32,46)(33,47)(34,48)(39,40)(41,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,39)(25,27)(28,43)(29,42)(30,45)(32,48)(33,47)(34,46)(38,40)(41,44);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,35,36,37)(24,40,30,41)(25,26,42,43)(31,46,47,48)(38,39,44,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6924353);