Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | |
1827 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)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,36,49)(24,29,40,47)(25,30,26,41)(28,34,43,35)(33,48,37,52)(38,45,39,46)(42,51,44,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,27,36,26,35)(22,28,41,49,31,43,30,32)(24,37,39,50,40,33,38,51)(29,44,45,48,47,42,46,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,42)(30,46)(31,47)(32,48)(34,51)(35,50)(36,37)(41,45)(43,44)(49,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,32,43,49)(29,45,47,46)(33,50,37,51)(42,52,44,48);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,39,40,38)(28,32,43,49)(29,46,47,45)(33,51,37,50)(42,48,44,52)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6471790);