Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,30)(29,42)(31,38)(32,40)(33,41)(34,46)(35,39)(37,43)(44,47)(45,48);(9,10)(11,16)(12,14)(13,15)(17,19,22,30,23,31,21,35)(18,24,26,28,27,39,36,38)(20,29,34,43,33,44,32,45)(25,37,40,42,41,48,46,47);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,29)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(28,42)(30,43)(31,44)(35,45)(36,46)(38,47)(39,48);(10,14)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,35)(20,32,33,34)(24,38,39,28)(25,40,41,46)(29,43,44,45)(37,47,48,42);(1,4,5,3)(2,6,7,8)(20,33)(25,41)(29,44)(32,34)(37,48)(40,46)(42,47)(43,45)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376520);