Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,30)(23,27,34,43)(24,35,37,28)(25,29,39,46)(31,40,33,42)(32,44,36,41)(38,49,50,45)(47,52,48,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,36)(25,31)(26,34)(28,44)(29,40)(30,43)(32,37)(33,39)(35,41)(38,48)(42,46)(45,52)(47,50)(49,51);(1,2,3,4)(23,31)(24,37)(25,39)(27,40)(28,35)(29,46)(32,48)(33,34)(36,47)(41,52)(42,43)(44,51);(6,11)(10,13)(12,19)(18,20)(21,24,26,37)(22,28,30,35)(23,32,34,36)(25,38,39,50)(27,41,43,44)(29,45,46,49)(31,47,33,48)(40,51,42,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,31)(25,39)(27,40)(29,46)(32,47)(33,34)(36,48)(38,50)(41,51)(42,43)(44,52)(45,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592921);