Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #20723 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
732 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)(3,13,14,39)(4,10)(5,11)(6,24,25,9)(7,12)(8,16,29,40)(15,37,18,45)(17,36)(19,33)(20,50,51,30)(21,34)(22,52,53,31)(23,35)(26,54)(27,42,32,46)(28,44)(38,41)(43,59)(47,60)(48,63,64,57)(49,58)(55,62)(56,61);(1,3)(2,8,12,29)(4,18)(5,16)(6,17)(7,14)(9,28,24,26)(10,32,34,27)(11,13,35,39)(15,21)(19,46)(20,47)(22,44)(23,40)(25,41)(30,55,50,56)(31,38,52,36)(33,45,58,37)(42,49)(43,51)(48,62)(53,54)(57,59,63,60)(61,64);(2,9)(3,14)(8,26)(10,30)(11,31)(12,24)(13,36)(15,18)(16,40)(17,41)(27,55)(28,29)(32,56)(33,57)(34,50)(35,52)(37,59)(38,39)(42,46)(43,47)(44,54)(45,60)(58,63)(61,62);(1,4,5,19,7,21,23,49)(2,10,11,33,12,34,35,58)(3,15,16,42,14,18,40,46)(6,20,22,48,25,51,53,64)(8,27,13,37,29,32,39,45)(9,30,31,57,24,50,52,63)(17,43,44,61,41,47,54,62)(26,55,36,59,28,56,38,60)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di,g^b=gi,g^c=gh,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401216);