Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14725 |
| 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 | |
2107 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)(3,14)(4,19)(5,24,27,11)(6,28)(7,13)(8,12)(9,17)(10,23)(15,41)(16,45,47,40)(18,34)(20,31)(21,48,51,36)(22,37)(25,39,26,53)(29,33)(30,38)(32,46,52,54)(35,44)(42,61,63,58)(43,57)(49,56,62,64)(50,59,55,60);(1,3,6,17,7,18,8,29)(2,9,12,14,13,33,28,34)(4,20,22,15,23,43,30,44)(5,16,25,46,27,47,26,54)(10,35,37,57,19,41,38,31)(11,32,39,40,24,52,53,45)(21,49,50,42,51,62,55,63)(36,58,59,64,48,61,60,56);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,30)(9,31)(11,36)(12,37)(13,19)(14,35)(16,42)(17,43)(18,44)(20,29)(24,48)(25,50)(26,55)(27,51)(28,38)(32,56)(33,57)(34,41)(39,59)(40,58)(45,61)(46,62)(47,63)(49,54)(52,64)(53,60);(1,5)(2,11)(3,16)(4,21)(6,25)(7,27)(8,26)(9,32)(10,36)(12,39)(13,24)(14,40)(15,42)(17,46)(18,47)(19,48)(20,49)(22,50)(23,51)(28,53)(29,54)(30,55)(31,56)(33,52)(34,45)(35,58)(37,59)(38,60)(41,61)(43,62)(44,63)(57,64)Polycyclic presentation:
2^9|a^2=fi,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,g^c=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, 302755);