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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1530 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)(20,25,35,42)(21,36,23,26)(29,45,46,38)(30,39,32,44)(33,48,47,40)(34,43,37,41);(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)(21,32,23,30)(25,40,42,48)(26,44,36,39)(29,37,46,34)(38,41,45,43);(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,41)(27,42)(28,45)(30,33)(31,46)(32,47)(36,43)(39,48)(40,44);(2,5)(6,8)(11,16)(13,15)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,44)(25,41,42,43)(29,33,46,47)(38,48,45,40);(1,3,4,7)(2,6,5,8)(18,27)(20,35)(24,28)(26,36)(29,46)(33,47)(34,37)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419343);