Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,32)(23,28,35,43)(24,29,34,42)(25,30,39,47)(26,31,36,44)(33,41,50,52)(37,45,38,46)(40,48,49,51);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,35)(22,28,32,43)(24,36,34,26)(25,38,39,37)(29,44,42,31)(30,46,47,45)(33,49,50,40)(41,51,52,48);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,26,23,27,34,36,35)(22,29,31,28,32,42,44,43)(25,37,40,33,39,38,49,50)(30,45,48,41,47,46,51,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(25,39)(30,47)(33,50)(37,38)(40,49)(41,52)(45,46)(48,51);(7,17)(11,14)(15,16)(19,20)(21,25,26,40,27,39,36,49)(22,30,31,48,32,47,44,51)(23,33,34,38,35,50,24,37)(28,41,42,46,43,52,29,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722981);