Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636908 |
| 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 | #24397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,33)(21,28)(22,26)(23,27)(24,31)(25,34)(30,44)(32,39)(35,42)(36,41)(37,40)(38,46)(43,47)(45,48);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,41)(28,39)(33,48)(34,47)(35,45)(37,46)(40,43)(42,44);(11,15)(13,16)(17,20,21,37,22,35,23,34)(18,25,27,42,26,40,28,33)(19,30,36,46,31,45,32,47)(24,38,41,44,29,43,39,48);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,26)(19,31)(25,40)(27,28)(30,45)(32,36)(33,42)(46,47);(2,5)(6,7)(10,14)(13,16)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)Polycyclic presentation:
2^9|c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544662);