Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1784 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,30,27,31,25,42)(23,32,36,45,37,34,35,28)(24,29,40,48,41,39,43,47)(33,49,50,46,38,51,52,44);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,37)(22,28,31,45)(24,38,41,33)(25,36,26,35)(29,46,39,44)(30,34,42,32)(40,50,43,52)(47,51,48,49);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,29)(23,33)(25,43)(26,40)(27,41)(28,44)(30,48)(31,39)(32,49)(34,51)(35,52)(36,50)(37,38)(42,47)(45,46);(8,16)(11,20)(13,15)(17,19)(21,22,25,42,27,31,26,30)(23,34,35,45,37,32,36,28)(24,39,43,48,41,29,40,47)(33,49,52,44,38,51,50,46);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,30,31,42)(23,35,37,36)(24,40,41,43)(28,32,45,34)(29,47,39,48)(33,50,38,52)(44,51,46,49)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,d^2=fi,e^2=i,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6942218);