Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 | #28325 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2031 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,26,29)(23,31,32,46)(24,28,36,45)(25,38,39,30)(27,40,41,34)(33,47,35,50)(37,52,48,44)(42,51,43,49);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,34,26,32,39,40)(22,27,30,31,29,41,38,46)(24,35,37,51,36,33,48,49)(28,43,44,50,45,42,52,47);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(23,32)(27,41)(31,46)(33,35)(34,40)(42,43)(47,50)(49,51);(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,42)(29,45)(30,44)(31,47)(32,35)(34,49)(38,52)(39,48)(40,51)(41,43)(46,50);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,29)(24,36)(27,41)(30,38)(31,46)(33,35)(37,48)(49,51)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405976);