Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,32)(24,29)(25,30)(26,40)(27,31)(28,34)(33,46)(35,42)(36,44)(37,45)(38,50)(39,43)(41,47)(48,51)(49,52);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,34,27,35,25,39)(22,28,30,32,31,43,40,42)(24,33,38,47,37,48,36,49)(29,41,44,46,45,52,50,51);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,29)(23,33)(25,36)(26,38)(27,37)(28,41)(30,44)(31,45)(32,46)(34,47)(35,48)(39,49)(40,50)(42,51)(43,52);(8,16)(11,20)(13,15)(17,19)(21,25,27,26)(22,30,31,40)(23,34,35,39)(24,36,37,38)(28,42,43,32)(29,44,45,50)(33,47,48,49)(41,51,52,46);(7,14)(10,18)(13,15)(17,19)(24,37)(29,45)(33,48)(36,38)(41,52)(44,50)(46,51)(47,49)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6416055);