Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 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)(21,22,27,40,26,30,25,31)(23,32,39,28,34,47,35,43)(24,29,41,52,37,45,38,46)(33,48,51,42,36,50,49,44);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,36)(25,39)(26,34)(27,35)(29,44)(30,43)(31,47)(32,40)(33,37)(38,49)(41,51)(42,45)(46,48)(50,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,37)(22,29,30,45)(23,33,34,36)(25,38,27,41)(28,42,43,44)(31,46,40,52)(32,48,47,50)(35,49,39,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,37)(29,45)(33,36)(38,41)(42,44)(46,52)(48,50)(49,51);(6,12)(10,13)(11,20)(18,19)(22,30)(23,34)(24,37)(31,40)(35,39)(38,41)(42,44)(48,50)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436619);