Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2700 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)(21,22,27,31)(23,32,35,42)(24,29,37,45)(25,30,26,39)(28,40,43,34)(33,47,50,52)(36,44,38,46)(41,48,51,49);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,50)(38,49)(44,47)(45,51)(46,52);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,27,37)(22,29,31,45)(23,33,35,50)(25,38,26,36)(28,41,43,51)(30,46,39,44)(32,47,42,52)(34,49,40,48);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,36,37,38)(28,42,43,32)(29,44,45,46)(33,48,50,49)(41,47,51,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,31)(28,43)(29,45)(30,39)(32,42)(41,51)(44,46)(47,52)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442305);