Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | #37544 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2049 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)(23,33)(24,38)(25,30)(26,31)(27,32)(28,35)(29,41)(34,42)(36,44)(37,43)(39,45)(40,46)(47,50)(48,51)(49,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,35,27,37,26,36)(22,28,31,33,32,44,30,43)(24,34,39,48,40,49,41,50)(29,42,46,47,45,52,38,51);(6,11)(7,13)(12,15)(14,19)(21,24,26,41,27,40,25,39)(22,29,30,38,32,45,31,46)(23,34,36,50,37,49,35,48)(28,42,43,51,44,52,33,47);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,38,45,46)(34,48,49,50)(42,51,52,47);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,40)(29,45)(34,49)(38,46)(39,41)(42,52)(47,51)(48,50)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406061);