Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42797 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,32)(23,28,34,42)(24,29,36,44)(25,30,39,47)(26,31,40,48)(33,41,50,52)(35,43,37,45)(38,46,49,51);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,34)(22,28,32,42)(24,35,36,37)(25,26,39,40)(29,43,44,45)(30,31,47,48)(33,49,50,38)(41,51,52,46);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,26,38,27,36,40,49)(22,29,31,46,32,44,48,51)(23,33,25,35,34,50,39,37)(28,41,30,43,42,52,47,45);(6,12)(10,13)(11,18)(17,19)(24,36)(29,44)(33,50)(35,37)(38,49)(41,52)(43,45)(46,51);(7,15)(10,13)(14,20)(17,19)(21,25,26,34,27,39,40,23)(22,30,31,42,32,47,48,28)(24,37,38,33,36,35,49,50)(29,45,46,41,44,43,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714509);