Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,27,31)(23,32,36,46)(24,29,37,43)(25,30,26,40)(28,34,42,35)(33,47,48,52)(38,44,39,45)(41,50,51,49);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(24,37)(29,43)(33,48)(38,39)(41,51)(44,45)(47,52)(49,50);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,27,36)(22,28,31,42)(24,33,37,48)(25,35,26,34)(29,41,43,51)(30,46,40,32)(38,50,39,49)(44,47,45,52);(6,13)(11,15)(12,14)(19,20)(21,24,27,37)(22,29,31,43)(23,33,36,48)(25,39,26,38)(28,41,42,51)(30,45,40,44)(32,47,46,52)(34,50,35,49);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,46)(29,44,43,45)(33,49,48,50)(41,52,51,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582423);