Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,46,27,32,26,33)(23,34,44,29,37,50,38,48)(24,39,43,47,36,30,41,52)(25,42,40,49,45,31,35,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,40)(25,43)(26,44)(27,37)(28,38)(30,42)(31,39)(32,48)(33,50)(34,46)(35,36)(41,45)(47,49)(51,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,44)(26,41)(27,36)(28,43)(29,31)(32,39)(33,47)(34,49)(37,40)(38,45)(42,48)(46,52)(50,51);(7,14)(10,13)(15,20)(18,19)(21,25,28,40,27,45,26,35)(22,31,46,51,32,42,33,49)(23,36,44,41,37,24,38,43)(29,47,50,30,48,52,34,39);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,32)(23,37)(24,36)(31,42)(33,46)(38,44)(41,43)(49,51)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088120);