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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 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)(23,33)(24,29)(25,31)(26,30)(27,32)(28,35)(34,48)(36,42)(37,45)(38,43)(39,44)(40,47)(41,49)(46,50)(51,52);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,35)(22,28,30,33)(24,37,36,27)(25,38,39,50)(29,32,42,45)(31,46,44,43)(34,40,49,51)(41,47,48,52);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,27,23,26,36,37,35)(22,29,32,33,30,42,45,28)(25,38,40,34,39,50,51,49)(31,43,47,48,44,46,52,41);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(22,30)(25,39)(28,33)(29,42)(32,45)(34,49)(38,50)(40,51);(7,17)(11,14)(15,16)(19,20)(21,25,37,51,26,39,27,40)(22,31,45,52,30,44,32,47)(23,34,24,38,35,49,36,50)(28,41,42,46,33,48,29,43)Polycyclic presentation:
2^9|b^2=i,c^2=fghi,d^2=fi,e^2=hi,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, 6722972);