Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4484 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)(3,4)(5,6,11,17,12,16,10,18)(7,19,22,13,24,37,23,36)(8,25,27,14,29,40,28,39)(9,15,31,44,33,42,32,43)(20,34,49,45,50,61,48,53)(21,46,51,35,52,41,30,59)(26,54,57,38,58,64,56,63)(47,60,55,65,67,68,66,62);(5,7)(6,13)(8,20)(9,30)(10,23)(11,22)(12,24)(14,34)(15,41)(16,36)(17,37)(18,19)(21,31)(25,53)(26,55)(27,49)(28,48)(29,50)(32,52)(33,51)(35,43)(38,62)(39,61)(40,45)(42,46)(44,59)(47,56)(54,68)(57,67)(58,66)(60,64)(63,65);(2,4)(5,8,12,29)(6,14,16,39)(7,20,24,50)(9,26,33,58)(10,27,11,28)(13,34,36,61)(15,38,42,63)(17,25,18,40)(19,45,37,53)(21,47,52,67)(22,48,23,49)(30,55,51,66)(31,56,32,57)(35,60,59,68)(41,62,46,65)(43,64,44,54);(5,9)(6,15)(7,21)(8,26)(10,32)(11,31)(12,33)(13,35)(14,38)(16,42)(17,44)(18,43)(19,46)(20,47)(22,51)(23,30)(24,52)(25,54)(27,57)(28,56)(29,58)(34,60)(36,59)(37,41)(39,63)(40,64)(45,65)(48,66)(49,55)(50,67)(53,62)(61,68);(6,16)(13,36)(14,39)(15,42)(17,18)(19,37)(25,40)(34,61)(35,59)(38,63)(41,46)(43,44)(45,53)(54,64)(60,68)(62,65)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,g^c=gi,h^c=hi,f^d=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286404);