Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,26,31,27,32,25,28)(23,33,36,43,34,29,42,45)(24,37,38,44,41,30,40,48)(35,46,50,52,39,47,49,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,34)(24,38)(25,26)(29,45)(31,32)(33,43)(35,49)(39,50)(40,41)(44,48)(46,47);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,27,34)(22,29,32,33)(24,39,41,35)(25,42,26,36)(28,43,31,45)(30,47,37,46)(38,49,40,50)(44,52,48,51);(7,14)(10,13)(15,20)(18,19)(21,24,25,40,27,41,26,38)(22,30,28,44,32,37,31,48)(23,35,42,49,34,39,36,50)(29,46,43,51,33,47,45,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,28)(23,36,34,42)(24,40,41,38)(29,43,33,45)(30,48,37,44)(35,50,39,49)(46,51,47,52)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093875);