Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2746 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)(21,22,26,31,27,32,25,30)(23,33,37,41,34,43,36,28)(24,29,39,45,38,44,40,46)(35,47,49,51,48,52,50,42);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,34)(22,28,32,41)(24,35,38,48)(25,37,26,36)(29,42,44,51)(30,33,31,43)(39,50,40,49)(45,52,46,47);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(23,34)(24,38)(28,41)(29,44)(33,43)(36,37)(39,40)(45,46);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,27,38)(22,29,32,44)(23,35,34,48)(25,40,26,39)(28,42,41,51)(30,46,31,45)(33,47,43,52)(36,50,37,49);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,34,37)(24,39,38,40)(28,43,41,33)(29,45,44,46)(35,49,48,50)(42,47,51,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436797);