Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #406018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14922 |
| 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 | |
987 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)(3,13,16,8)(4,9)(5,22,23,52)(6,12)(7,11)(10,24,33,53)(14,27,18,36)(15,41)(17,38,44,29)(19,47,48,51)(20,32)(21,31)(25,35)(26,34)(28,43)(30,49,58,54)(37,60)(39,61)(40,56,45,57)(42,50)(46,64)(55,63)(59,62);(1,3,6,17,7,16,26,44)(2,8,12,29,11,13,34,38)(4,18,20,45,21,14,25,40)(5,15,24,43,23,42,53,60)(9,27,32,56,31,36,35,57)(10,28,22,50,33,37,52,41)(19,46,49,63,48,39,54,62)(30,55,47,61,58,59,51,64);(1,4,7,21)(2,9,11,31)(3,14,16,18)(5,19,23,48)(6,25,26,20)(8,27,13,36)(10,30,33,58)(12,35,34,32)(15,39,42,46)(17,45,44,40)(22,51,52,47)(24,54,53,49)(28,55,37,59)(29,57,38,56)(41,61,50,64)(43,63,60,62);(1,5)(2,10,11,33)(3,15)(4,19)(6,24)(7,23)(8,28,13,37)(9,30,31,58)(12,22,34,52)(14,39)(16,42)(17,43)(18,46)(20,49)(21,48)(25,54)(26,53)(27,55,36,59)(29,50,38,41)(32,47,35,51)(40,62)(44,60)(45,63)(56,61,57,64)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313215);