Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,37)(3,13,17,42,16,8,28,31)(4,18,22,38,21,9,25,34)(5,10,24,36,23,35,26,39)(14,40,19,49,44,29,47,32)(15,30,46,57,45,41,48,58)(20,50,53,62,54,33,55,61)(43,56,52,60,64,63,51,59);(1,3)(2,8,11,13)(4,19,21,47)(5,15,23,45)(6,17)(7,16)(9,32)(10,30)(12,31,37,42)(14,22,44,25)(18,49)(20,52)(24,46,26,48)(27,28)(29,38)(33,60,50,59)(34,40)(35,41)(36,57)(39,58)(43,55)(51,54)(53,64)(56,61,63,62);(1,4,7,21)(2,9,11,18)(3,14,16,44)(5,20,23,54)(6,25,27,22)(8,29,13,40)(10,33,35,50)(12,38,37,34)(15,43,45,64)(17,47,28,19)(24,55,26,53)(30,56,41,63)(31,49,42,32)(36,62,39,61)(46,51,48,52)(57,59,58,60);(1,5)(2,10,11,35)(3,15,16,45)(4,20)(6,26)(7,23)(8,30)(9,33,18,50)(12,39,37,36)(13,41)(14,43,44,64)(17,48,28,46)(19,51,47,52)(21,54)(22,55)(24,27)(25,53)(29,56)(31,58)(32,59)(34,62,38,61)(40,63)(42,57)(49,60)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343705);