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×#548 | 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)(23,32)(24,29)(25,30)(26,41)(27,31)(28,34)(33,48)(35,43)(36,50)(37,46)(38,47)(39,52)(40,44)(42,49)(45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34,27,35,25,40)(22,28,30,32,31,44,41,43)(24,36,39,51,37,33,38,49)(29,45,47,50,46,42,52,48);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24,27,37)(22,29,31,46)(23,33,35,36)(25,38,26,39)(28,42,44,45)(30,47,41,52)(32,48,43,50)(34,49,40,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,37)(29,46)(33,36)(38,39)(42,45)(47,52)(48,50)(49,51);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,34,35,40)(24,38,37,39)(28,43,44,32)(29,47,46,52)(33,49,36,51)(42,50,45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,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, 6477038);