Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| 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 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,7,14,33,50,9,16,36,53,6,13,32,49)(3,17,19,54,18,40,55,51,20,48,34,61,31,10,37,39)(4,21,24,56,25,11,41,43,26,44,35,62,23,38,58,52)(5,27,29,60,22,42,57,63,30,12,45,47,28,46,59,64);(1,3)(2,10)(4,22)(5,23)(6,31)(7,18)(8,34)(9,20)(11,42)(12,44)(13,48)(14,17)(15,51)(16,40)(19,36)(21,27)(24,59)(25,30)(26,28)(29,41)(32,55)(33,37)(35,57)(38,46)(39,53)(43,64)(45,58)(47,56)(49,54)(50,61)(52,63)(60,62);(1,4)(2,11)(3,5)(6,23)(7,25)(8,35)(9,26)(10,27)(12,40)(13,21)(14,44)(15,52)(16,38)(17,42)(18,22)(19,45)(20,30)(24,36)(28,31)(29,34)(32,41)(33,58)(37,57)(39,47)(43,53)(46,48)(49,62)(50,56)(51,60)(54,64)(55,59)(61,63);(1,5)(2,12)(3,4)(6,28)(7,22)(8,29)(9,30)(10,38)(11,40)(13,42)(14,46)(15,47)(16,27)(17,21)(18,25)(19,24)(20,26)(23,31)(32,59)(33,57)(34,35)(36,45)(37,58)(39,52)(41,55)(43,51)(44,48)(49,63)(50,64)(53,60)(54,56)(61,62)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=defi,c^b=cefh,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405906);