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 | |
1535 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)(17,18,22,27)(19,28,31,24)(20,32,35,25)(21,37,23,26)(29,39,46,45)(30,40,38,41)(33,42,47,48)(34,43,36,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,28)(20,33,35,47)(21,30,23,38)(25,42,32,48)(26,40,37,41)(29,34,46,36)(39,44,45,43);(10,14)(13,15)(17,20,22,35)(18,25,27,32)(19,29,31,46)(21,36,23,34)(24,39,28,45)(26,44,37,43)(30,47,38,33)(40,42,41,48);(2,5)(6,8)(11,16)(13,15)(17,19)(18,24)(20,29)(21,38)(22,31)(23,30)(25,39)(26,41)(27,28)(32,45)(33,36)(34,47)(35,46)(37,40)(42,43)(44,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,37)(19,30,31,38)(20,34,35,36)(24,40,28,41)(25,43,32,44)(29,33,46,47)(39,48,45,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7419633);