Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15278 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1607 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)(3,15)(4,21)(5,11,29,40)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,46,47,38)(19,34)(20,33)(22,50)(23,52,55,49)(24,37)(26,36)(27,57,48,32)(28,41,30,39)(35,56,62,54)(42,45)(43,59,51,64)(44,60)(53,61,58,63);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,31)(11,38)(12,33)(13,15)(14,34)(16,26)(17,28)(21,42)(23,53)(24,44)(25,45)(29,48)(30,47)(32,39)(35,61)(36,60)(37,50)(40,46)(41,57)(43,56)(49,64)(51,54)(52,59)(55,58)(62,63);(1,4,7,25,8,26,6,24)(2,10,13,21,14,37,12,36)(3,16,19,45,20,22,18,44)(5,23,30,56,29,55,28,54)(9,31,33,60,34,50,15,42)(11,35,41,49,40,62,39,52)(17,43,48,53,47,51,27,58)(32,59,46,63,57,64,38,61);(1,5,6,28,8,29,7,30)(2,11,12,39,14,40,13,41)(3,17,18,27,20,47,19,48)(4,23,24,54,26,55,25,56)(9,32,15,38,34,57,33,46)(10,35,36,52,37,62,21,49)(16,43,44,58,22,51,45,53)(31,59,42,61,50,64,60,63)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327684);