Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15277 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 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)(3,14,18,34)(4,10)(5,24)(6,12)(7,29)(8,13)(9,17,33,28)(11,26)(15,42,19,41)(16,45,46,32)(20,50)(21,36)(22,37)(23,54)(25,56,47,38)(27,39)(30,40)(31,49,59,44)(35,52)(43,58,48,64)(51,63,57,60)(53,61)(55,62);(1,3)(2,9)(4,19)(5,25)(6,28)(7,17)(8,18)(10,31)(11,38)(12,34)(13,33)(14,29)(15,22)(16,26)(20,51)(21,44)(23,49)(24,45)(27,47)(30,46)(32,39)(35,60)(36,42)(37,59)(40,56)(41,54)(43,55)(48,52)(50,64)(53,57)(58,61)(62,63);(1,4,6,21,8,22,7,23)(2,10,12,36,13,37,29,54)(3,15,17,44,18,19,28,49)(5,20,26,52,27,53,30,55)(9,31,14,41,33,59,34,42)(11,35,39,61,40,62,24,50)(16,43,25,57,46,48,47,51)(32,58,38,60,45,64,56,63);(1,5,6,26,8,27,7,30)(2,11,12,39,13,40,29,24)(3,16,17,25,18,46,28,47)(4,20,21,52,22,53,23,55)(9,32,14,38,33,45,34,56)(10,35,36,61,37,62,54,50)(15,43,44,57,19,48,49,51)(31,58,41,60,59,64,42,63)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327569);