Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,25,29,26,30,36,42)(23,27,33,39,32,38,45,49)(24,28,35,41,34,40,46,50)(31,37,44,48,43,47,51,52);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,37)(29,39)(30,38)(34,43)(35,44)(36,45)(40,47)(41,48)(42,49)(46,51)(50,52);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24)(22,28)(23,31)(25,35)(26,34)(27,37)(29,41)(30,40)(32,43)(33,44)(36,46)(38,47)(39,48)(42,50)(45,51)(49,52);(7,16)(11,14)(15,17)(19,20)(23,32)(24,34)(27,38)(28,40)(33,45)(35,46)(39,49)(41,50);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294126);