Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| 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 | |
1749 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,31)(23,32,36,47)(24,29,38,44)(25,30,26,40)(28,34,42,35)(33,46,49,51)(37,43,39,45)(41,50,52,48);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,47)(24,33,37,48,38,49,39,50)(29,41,45,46,44,52,43,51);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,33)(25,39)(26,37)(27,38)(28,41)(30,45)(31,44)(32,46)(34,50)(35,48)(36,49)(40,43)(42,52)(47,51);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,44,45)(33,48,49,50)(41,51,52,46);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,38)(29,44)(33,49)(37,39)(41,52)(43,45)(46,51)(48,50)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403687);