Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15567 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,33,30,34)(4,20,23,37,24,41,22,10)(5,25,27,40,29,39,28,11)(16,35,44,48,46,31,45,42)(17,32,58,38,47,55,26,56)(21,50,52,49,54,36,53,62)(43,61,57,59,63,64,51,60);(1,3,7,18,8,19,6,30)(2,9,12,33,14,34,13,15)(4,16,23,44,24,46,22,45)(5,26,28,47,29,58,27,17)(10,35,20,48,37,31,41,42)(11,38,39,32,40,56,25,55)(21,51,53,63,54,57,52,43)(36,61,49,60,50,64,62,59);(1,4,8,24)(2,10,14,37)(3,16,19,46)(5,21,29,54)(6,23,7,22)(9,31,34,35)(11,36,40,50)(12,41,13,20)(15,42,33,48)(17,43,47,63)(18,45,30,44)(25,49,39,62)(26,57,58,51)(27,53,28,52)(32,59,55,60)(38,61,56,64);(1,5)(2,11,14,40)(3,17,19,47)(4,21)(6,27)(7,28)(8,29)(9,32)(10,36,37,50)(12,39,13,25)(15,38)(16,43,46,63)(18,26,30,58)(20,49,41,62)(22,52)(23,53)(24,54)(31,59)(33,56)(34,55)(35,60)(42,61)(44,51,45,57)(48,64)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343901);