Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21107 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,14,31,41,9,15,30,42,7,13,5,28)(3,16,20,45,21,36,51,60,22,43,18,34,19,10,29,35)(4,23,26,53,25,11,37,38,27,39,56,62,32,40,24,54)(17,33,49,58,48,44,61,63,50,59,46,64,47,52,55,57);(1,3,8,20,6,21,31,51,9,22,30,18,7,19,5,29)(2,10,12,35,14,16,41,45,15,36,42,60,13,43,28,34)(4,17,26,49,25,48,37,61,27,50,56,46,32,47,24,55)(11,33,38,58,39,44,62,63,40,59,54,64,23,52,53,57);(1,4)(2,11)(3,17)(5,24)(6,25)(7,32)(8,26)(9,27)(10,33)(12,38)(13,23)(14,39)(15,40)(16,44)(18,46)(19,47)(20,49)(21,48)(22,50)(28,53)(29,55)(30,56)(31,37)(34,57)(35,58)(36,59)(41,62)(42,54)(43,52)(45,63)(51,61)(60,64);(1,5,6,8,9,31,7,30)(2,12,13,28,15,42,14,41)(3,18,19,51,22,20,21,29)(4,24,25,26,27,37,32,56)(10,34,16,35,36,45,43,60)(11,38,23,53,40,54,39,62)(17,46,47,61,50,49,48,55)(33,57,44,58,59,63,52,64)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=ei,f^a=fi,d^b=dh,e^b=ei,f^c=fi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402154);