Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460625 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,29,8,12,6,13)(3,9,18,33,16,34,17,32)(4,19,23,10,24,41,22,37)(5,25,26,11,28,40,27,39)(14,42,45,30,44,35,20,51)(15,46,49,59,48,60,50,31)(21,36,54,52,56,38,55,57)(43,63,53,61,47,58,64,62);(1,3,6,18,8,16,7,17)(2,9,29,32,12,34,13,33)(4,20,22,14,24,45,23,44)(5,15,26,50,28,48,27,49)(10,35,41,30,37,42,19,51)(11,31,25,46,39,59,40,60)(21,53,54,43,56,64,55,47)(36,61,57,58,38,62,52,63);(1,4,8,24)(2,10,12,37)(3,14,16,44)(5,21,28,56)(6,23,7,22)(9,30,34,51)(11,38,39,36)(13,41,29,19)(15,47,48,43)(17,45,18,20)(25,57,40,52)(26,55,27,54)(31,58,59,63)(32,35,33,42)(46,61,60,62)(49,53,50,64);(1,5)(2,11)(3,15)(4,21)(6,26)(7,27)(8,28)(9,31)(10,36)(12,39)(13,40)(14,43)(16,48)(17,49)(18,50)(19,52)(20,53)(22,54)(23,55)(24,56)(25,29)(30,58)(32,46)(33,60)(34,59)(35,61)(37,38)(41,57)(42,62)(44,47)(45,64)(51,63)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367822);