Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1537 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)(9,10,12,14)(11,15,16,13)(17,18,22,28)(19,26,31,24)(20,25,34,44)(21,35,23,27)(29,42,47,38)(30,39,36,40)(32,48,46,41)(33,45,37,43);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,28,26)(20,32,34,46)(21,30,23,36)(25,41,44,48)(27,39,35,40)(29,37,47,33)(38,43,42,45);(10,14)(13,15)(17,20,22,34)(18,25,28,44)(19,29,31,47)(21,33,23,37)(24,38,26,42)(27,43,35,45)(30,46,36,32)(39,41,40,48);(2,5)(6,8)(11,16)(13,15)(17,19)(18,26)(20,29)(21,36)(22,31)(23,30)(24,28)(25,42)(27,39)(32,33)(34,47)(35,40)(37,46)(38,44)(41,43)(45,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,28,35)(19,30,31,36)(20,33,34,37)(24,39,26,40)(25,43,44,45)(29,46,47,32)(38,41,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419568);