Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2788 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,26,29)(23,31,32,46)(24,28,36,45)(25,38,40,30)(27,39,41,34)(33,47,35,49)(37,52,51,44)(42,50,43,48);(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,26,32)(22,27,29,41)(24,35,36,33)(25,39,40,34)(28,43,45,42)(30,46,38,31)(37,48,51,50)(44,47,52,49);(2,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(23,32)(27,41)(31,46)(33,35)(34,39)(42,43)(47,49)(48,50);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,24)(22,28)(23,33)(25,37)(26,36)(27,42)(29,45)(30,44)(31,47)(32,35)(34,48)(38,52)(39,50)(40,51)(41,43)(46,49);(6,14)(11,20)(12,13)(18,19)(22,29)(24,36)(27,41)(30,38)(31,46)(33,35)(37,51)(48,50)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396780);