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 | |
1788 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)(21,22,27,31,28,32,26,25)(23,33,36,45,37,48,35,29)(24,30,40,39,41,47,43,42)(34,49,50,46,38,52,51,44);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,37)(22,29,32,45)(24,38,41,34)(25,33,31,48)(26,36,27,35)(30,46,47,44)(39,49,42,52)(40,50,43,51);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,30)(23,34)(25,42)(26,43)(27,40)(28,41)(29,44)(31,39)(32,47)(33,49)(35,51)(36,50)(37,38)(45,46)(48,52);(8,16)(11,20)(13,15)(17,19)(21,25,26,32,28,31,27,22)(23,29,35,48,37,45,36,33)(24,39,43,30,41,42,40,47)(34,46,51,49,38,44,50,52);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,31,32,25)(23,35,37,36)(24,40,41,43)(29,33,45,48)(30,42,47,39)(34,50,38,51)(44,52,46,49)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,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, 6942090);