Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,31)(23,32,35,43)(24,29,38,48)(25,30,26,41)(28,39,44,34)(33,49,36,52)(37,46,40,47)(42,50,45,51);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18);(2,4)(21,23,25,39,27,35,26,34)(22,28,41,43,31,44,30,32)(24,36,40,51,38,33,37,50)(29,45,46,49,48,42,47,52);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,33)(25,40)(26,37)(27,38)(28,42)(30,47)(31,48)(32,49)(34,51)(35,36)(39,50)(41,46)(43,52)(44,45);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,37,38,40)(28,43,44,32)(29,46,48,47)(33,50,36,51)(42,49,45,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6593261);