Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,29)(20,33)(21,26)(22,27)(23,38)(24,31)(25,34)(28,37)(30,43)(32,41)(35,40)(36,42)(39,48)(44,46)(45,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,39)(26,43)(27,29)(28,41)(33,40)(34,46)(36,48)(37,47)(38,45)(42,44);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,22)(23,36)(24,39)(26,28)(27,42)(29,44)(31,35)(32,48)(33,38)(34,37)(40,45)(41,43)(46,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,22,20)(18,26,27,33)(19,30,31,35)(23,36,37,34)(24,40,29,43)(25,38,42,28)(32,46,47,48)(39,41,44,45);(2,5)(6,8)(10,14)(13,15)(18,27)(20,21)(24,29)(28,38)(30,35)(34,36)(41,45)(46,48)Polycyclic presentation:
2^9|a^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7143873);