Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6,11,17,12,18,10,16)(7,19,23,13,24,36,22,37)(8,25,29,39,27,40,28,14)(9,15,32,43,33,44,31,42)(20,46,50,45,48,34,49,60)(21,41,51,35,52,61,30,58)(26,53,57,63,55,64,56,38)(47,65,67,62,66,59,54,68);(1,3,4,2)(5,7)(6,13)(8,20)(9,30)(10,23)(11,22)(12,24)(14,34)(15,41)(16,36)(17,19)(18,37)(21,31)(25,45)(26,54)(27,48)(28,50)(29,49)(32,52)(33,51)(35,42)(38,62)(39,46)(40,60)(43,58)(44,61)(47,56)(53,65)(55,67)(57,66)(59,64)(63,68);(5,8,10,28,12,27,11,29)(6,14,17,25,18,39,16,40)(7,20,23,50,24,48,22,49)(9,26,31,56,33,55,32,57)(13,34,19,45,37,46,36,60)(15,38,43,53,44,63,42,64)(21,47,51,67,52,66,30,54)(35,59,41,62,58,65,61,68);(5,9,11,32,12,33,10,31)(6,15,17,43,18,44,16,42)(7,21,23,51,24,52,22,30)(8,26,29,57,27,55,28,56)(13,35,36,61,37,58,19,41)(14,38,25,53,39,63,40,64)(20,47,50,67,48,66,49,54)(34,59,60,68,46,65,45,62);(8,27)(14,39)(20,48)(25,40)(26,55)(28,29)(34,46)(38,63)(45,60)(47,66)(49,50)(53,64)(54,67)(56,57)(59,65)(62,68)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290009);