Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,45)(30,39)(32,44)(33,48)(34,41)(35,42)(36,43)(38,46)(40,47);(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,27,39)(20,33,35,29)(21,31,22,30)(25,40,42,38)(26,44,37,28)(34,46,36,47)(41,48,43,45);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,43)(27,42)(28,45)(30,47)(31,46)(32,33)(37,41)(39,40)(44,48);(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,44)(25,41,42,43)(29,46,33,47)(38,48,40,45);(9,11,12,15)(10,14,13,16)(18,27)(24,39)(25,42)(26,37)(28,44)(38,40)(41,43)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402066);