Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2344 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13907 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,14)(3,15,18,33,19,9,29,34)(4,20,24,37,25,40,23,10)(5,26,27,38,28,11,30,39)(16,43,45,41,46,35,21,31)(17,32,48,58,49,42,47,59)(22,52,53,56,55,50,54,36)(44,60,51,61,64,62,63,57);(1,3,6,18,8,19,7,29)(2,9,14,34,12,15,13,33)(4,21,24,46,25,45,23,16)(5,17,27,47,28,49,30,48)(10,35,20,41,37,43,40,31)(11,32,39,59,26,42,38,58)(22,51,54,64,55,63,53,44)(36,60,50,61,56,62,52,57);(1,4,6,23,8,25,7,24)(2,10,13,20,12,37,14,40)(3,16,18,45,19,46,29,21)(5,22,27,53,28,55,30,54)(9,31,33,35,15,41,34,43)(11,36,38,50,26,56,39,52)(17,44,47,63,49,64,48,51)(32,57,58,60,42,61,59,62);(1,5)(2,11)(3,17)(4,22)(6,27)(7,30)(8,28)(9,32)(10,36)(12,26)(13,38)(14,39)(15,42)(16,44)(18,47)(19,49)(20,50)(21,51)(23,53)(24,54)(25,55)(29,48)(31,57)(33,58)(34,59)(35,60)(37,56)(40,52)(41,61)(43,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269291);