Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
1696 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,29,26,30,36,42)(23,27,33,39,31,37,44,48)(24,28,35,41,34,40,46,50)(32,38,45,49,43,47,51,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(23,31)(27,37)(32,43)(33,44)(38,47)(39,48)(45,51)(49,52);(2,4)(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,38)(29,39)(30,37)(34,43)(35,45)(36,44)(40,47)(41,49)(42,48)(46,51)(50,52);(6,12)(10,13)(11,20)(18,19)(23,31)(24,34)(27,37)(28,40)(33,44)(35,46)(39,48)(41,50);(21,24,25,35,26,34,36,46)(22,28,29,41,30,40,42,50)(23,32,33,45,31,43,44,51)(27,38,39,49,37,47,48,52)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285198);