Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470368 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,31)(23,32,35,42)(24,29,36,44)(25,30,26,40)(28,39,43,34)(33,46,47,52)(37,45,38,50)(41,49,51,48);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,25,39,27,35,26,34)(22,28,40,42,31,43,30,32)(24,33,37,49,36,47,38,48)(29,41,50,52,44,51,45,46);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,33)(25,37)(26,38)(27,36)(28,41)(30,45)(31,44)(32,46)(34,48)(35,47)(39,49)(40,50)(42,52)(43,51);(7,14)(10,18)(13,15)(17,19)(24,36)(29,44)(33,47)(37,38)(41,51)(45,50)(46,52)(48,49);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,30,31,40)(23,34,35,39)(24,37,36,38)(28,42,43,32)(29,45,44,50)(33,48,47,49)(41,52,51,46)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384909);