Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1527 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)(3,7,8,6)(11,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,43)(33,39)(34,40)(35,41)(37,44)(45,47)(46,48);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,33,44,34,45,35,46)(25,37,41,42,40,48,39,47);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,41)(27,40)(28,42)(30,46)(31,44)(32,45)(36,39)(38,48)(43,47);(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,39,40,41)(29,44,45,46)(37,47,48,42);(9,11,12,15)(10,14,13,16)(20,34)(25,40)(29,45)(33,35)(37,48)(39,41)(42,47)(44,46)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402367);