Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,36,43)(24,29,40,48)(25,30,26,41)(28,34,44,35)(33,49,37,52)(38,46,39,47)(42,51,45,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,44,41,43)(24,37,38,51,40,33,39,50)(29,45,47,52,48,42,46,49);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,42)(30,47)(31,48)(32,49)(34,51)(35,50)(36,37)(41,46)(43,52)(44,45);(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)(28,32,44,43)(29,46,48,47)(33,50,37,51)(42,52,45,49);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,35,36,34)(24,39,40,38)(28,43,44,32)(29,47,48,46)(33,50,37,51)(42,52,45,49)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477020);