Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23307 |
| prime radical | #3018 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,7,14,32,49,9,16,35,52,6,13,31,48)(3,17,19,53,18,38,54,50,20,47,33,60,30,10,36,37)(4,21,24,56,25,11,39,41,26,42,34,61,23,43,58,51)(5,12,28,45,22,40,57,63,29,46,55,62,27,44,59,64);(1,3)(2,10)(4,22)(5,23)(6,30)(7,18)(8,33)(9,20)(11,40)(12,21)(13,47)(14,17)(15,50)(16,38)(19,35)(24,59)(25,29)(26,27)(28,39)(31,54)(32,36)(34,57)(37,52)(41,64)(42,46)(43,44)(45,61)(48,53)(49,60)(51,63)(55,58)(56,62);(1,4)(2,11)(3,5)(6,23)(7,25)(8,34)(9,26)(10,12)(13,21)(14,42)(15,51)(16,43)(17,40)(18,22)(19,55)(20,29)(24,35)(27,30)(28,33)(31,39)(32,58)(36,57)(37,62)(38,46)(41,52)(44,47)(45,50)(48,61)(49,56)(53,64)(54,59)(60,63);(1,5)(2,12)(3,4)(6,27)(7,22)(8,28)(9,29)(10,11)(13,44)(14,40)(15,45)(16,46)(17,42)(18,25)(19,24)(20,26)(21,47)(23,30)(31,59)(32,57)(33,34)(35,55)(36,58)(37,41)(38,43)(39,54)(48,64)(49,63)(50,51)(52,62)(53,61)(56,60)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405905);