Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473852 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16884 |
| 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 | |
852 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)(3,14)(4,20)(5,25,28,40)(6,30)(7,13)(8,12)(9,17)(10,23)(11,29,37,27)(15,41)(16,32,49,61)(18,33)(19,38)(21,53,54,36)(22,35)(24,39)(26,52,55,34)(31,45)(42,50,60,48)(43,63,64,57)(44,51)(46,58)(47,59,56,62);(1,3,6,18,7,19,8,17)(2,9,30,14,13,33,12,38)(4,15,22,45,23,46,24,44)(5,16,29,48,28,49,27,50)(10,31,39,58,20,51,35,41)(11,32,25,42,37,61,40,60)(21,43,55,47,54,64,26,56)(34,57,36,62,52,63,53,59);(1,4,7,23)(2,10,13,20)(3,15,19,46)(5,26,28,55)(6,24,8,22)(9,31,33,51)(11,36,37,53)(12,39,30,35)(14,41,38,58)(16,47,49,56)(17,45,18,44)(21,29,54,27)(25,34,40,52)(32,59,61,62)(42,63,60,57)(43,50,64,48);(1,5)(2,11)(3,16)(4,21)(6,29)(7,28)(8,27)(9,32)(10,34)(12,40)(13,37)(14,42)(15,43)(17,50)(18,48)(19,49)(20,52)(22,55)(23,54)(24,26)(25,30)(31,57)(33,61)(35,53)(36,39)(38,60)(41,59)(44,56)(45,47)(46,64)(51,63)(58,62)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 381049);