Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28319 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,31)(24,28)(25,38)(26,29)(27,34)(30,39)(32,45)(33,46)(35,50)(36,43)(37,52)(40,48)(41,49)(42,51)(44,47);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,34,26,33,39,49)(22,27,30,31,29,41,38,46)(24,35,37,51,36,32,47,48)(28,42,44,50,43,40,52,45);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,26,36)(22,28,29,43)(23,32,33,35)(25,37,39,47)(27,40,41,42)(30,44,38,52)(31,45,46,50)(34,48,49,51);(7,13)(10,17)(14,19)(18,20)(23,33)(24,36)(27,41)(28,43)(31,46)(34,49)(37,47)(44,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,29)(27,41)(28,43)(30,38)(31,46)(40,42)(44,52)(45,50)Polycyclic presentation:
2^9|b^2=f,c^2=gh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405918);