Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,30)(23,31,34,45)(24,28,36,42)(25,38,32,29)(27,39,41,35)(33,44,47,52)(37,49,46,43)(40,50,51,48);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,32,39,26,34,25,35)(22,27,38,45,30,41,29,31)(24,33,46,50,36,47,37,48)(28,40,49,52,42,51,43,44);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,47)(35,48)(38,49)(39,50)(41,51)(45,52);(6,11)(10,16)(12,18)(17,20)(23,34)(24,36)(27,41)(28,42)(31,45)(35,39)(37,46)(43,49);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,34)(27,41)(31,45)(33,47)(35,39)(40,51)(44,52)(48,50)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,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, 6343960);