Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2720 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)(21,22,26,32,27,30,25,31)(23,33,35,42,36,46,40,28)(24,29,37,44,38,43,39,45)(34,47,49,51,50,52,48,41);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,36)(22,28,30,42)(24,34,38,50)(25,35,26,40)(29,41,43,51)(31,33,32,46)(37,48,39,49)(44,52,45,47);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,34)(25,39)(26,37)(27,38)(28,41)(30,43)(31,45)(32,44)(33,47)(35,49)(36,50)(40,48)(42,51)(46,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,30)(28,42)(29,43)(31,32)(33,46)(41,51)(44,45)(47,52);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,31,30,32)(23,35,36,40)(24,37,38,39)(28,33,42,46)(29,44,43,45)(34,48,50,49)(41,52,51,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436610);