Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2231 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)(3,6)(4,5)(7,8)(17,18)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)(31,42)(32,41)(33,40)(34,39)(35,48)(36,44)(37,45)(38,46)(43,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32,24,33,22,20)(18,25,28,40,30,41,29,26)(21,31,36,35,38,47,37,34)(27,39,45,43,46,48,44,42);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,24,32,23,19)(18,26,29,41,30,40,28,25)(21,35,37,31,38,34,36,47)(27,43,44,39,46,42,45,48);(11,16)(13,15)(17,21,24,38)(18,27,30,46)(19,31,33,47)(20,34,32,35)(22,37,23,36)(25,39,41,48)(26,42,40,43)(28,45,29,44);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,20,33,32)(21,36,38,37)(25,40,41,26)(27,44,46,45)(31,35,47,34)(39,42,48,43)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413684);