Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(11,15)(14,16)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,32)(25,35)(30,45)(31,40)(34,48)(36,44)(37,43)(38,42)(39,46)(41,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,34)(21,37)(22,32)(23,31)(25,41)(26,43)(27,29)(28,40)(30,36)(33,45)(35,46)(38,47)(39,42)(44,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,36)(18,25,27,42)(19,30,32,34)(21,35,23,38)(24,39,29,41)(26,33,28,44)(31,46,37,47)(40,45,43,48);(10,13)(14,16)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,43)(25,33,42,44)(30,46,34,47)(39,45,41,48);(9,11,12,15)(10,14,13,16)(18,27)(19,32)(25,42)(26,28)(30,34)(31,37)(33,44)(46,47)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6399412);