Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,33)(23,34,36,29)(24,30,39,48)(25,41,28,31)(26,32,42,44)(35,49,50,43)(37,47,51,45)(38,52,40,46);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,36)(22,29,33,34)(24,37,39,51)(25,26,28,42)(30,45,48,47)(31,32,41,44)(35,38,50,40)(43,52,49,46);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,25,38,27,39,28,40)(22,30,41,52,33,48,31,46)(23,35,42,51,36,50,26,37)(29,43,32,47,34,49,44,45);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,25,27,28)(22,31,33,41)(23,26,36,42)(24,38,39,40)(29,32,34,44)(30,46,48,52)(35,37,50,51)(43,47,49,45);(7,17)(11,14)(15,16)(19,20)(21,26,25,23,27,42,28,36)(22,32,41,34,33,44,31,29)(24,37,38,35,39,51,40,50)(30,47,52,49,48,45,46,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722982);