Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405282 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14911 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,36)(4,9,21,33)(5,10,23,35)(6,24,26,11)(8,25,29,16)(14,37,18,43)(15,38,42,58)(19,46,48,31)(20,49,50,32)(22,51,53,34)(27,45,30,40)(28,52,55,41)(39,60,44,59)(47,57,64,63)(54,61,56,62);(1,3)(2,8)(4,18,21,14)(5,15)(6,25)(7,17)(9,30,33,27)(10,28)(11,36)(12,29)(13,24)(16,26)(19,44,48,39)(20,40,50,45)(22,52)(23,42)(31,56,46,54)(32,37,49,43)(34,58)(35,55)(38,51)(41,53)(47,61,64,62)(57,59,63,60);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,27)(10,31)(11,32)(12,33)(13,37)(15,39)(16,40)(17,18)(22,47)(23,48)(24,49)(25,45)(26,50)(28,54)(29,30)(34,57)(35,46)(36,43)(38,59)(41,61)(42,44)(51,63)(52,62)(53,64)(55,56)(58,60);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,28)(9,31)(11,34)(12,35)(13,38)(14,39)(16,41)(17,42)(18,44)(20,47)(21,48)(24,51)(25,52)(26,53)(27,54)(29,55)(30,56)(32,57)(33,46)(36,58)(37,59)(40,61)(43,60)(45,62)(49,63)(50,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312479);