Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436971 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15578 |
| 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 | |
919 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)(3,14,19,43)(4,20)(5,11,29,38)(6,12)(7,30)(8,13)(9,17,33,18)(10,24)(15,31,21,51)(16,36)(22,53,57,50)(23,35)(25,39)(26,58)(27,37,28,40)(32,47)(34,55,61,56)(41,45,49,46)(42,48)(44,59)(52,64)(54,62)(60,63);(1,3,7,18,8,19,6,17)(2,9,12,14,13,33,30,43)(4,21,24,45,25,15,23,46)(5,26,28,16,29,47,27,48)(10,31,20,49,35,51,39,41)(11,36,37,58,38,42,40,32)(22,54,56,52,57,63,55,44)(34,60,53,64,61,62,50,59);(1,4)(2,10)(3,15)(5,22)(6,24)(7,23)(8,25)(9,31)(11,34)(12,39)(13,35)(14,41)(16,44)(17,46)(18,45)(19,21)(20,30)(26,54)(27,56)(28,55)(29,57)(32,59)(33,51)(36,62)(37,50)(38,61)(40,53)(42,60)(43,49)(47,63)(48,52)(58,64);(1,5)(2,11,13,38)(3,16)(4,22)(6,28)(7,27)(8,29)(9,32,33,58)(10,34,35,61)(12,40,30,37)(14,42,43,36)(15,44)(17,47)(18,26)(19,48)(20,50,39,53)(21,52)(23,56)(24,55)(25,57)(31,59,51,64)(41,60,49,62)(45,54)(46,63)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72287080, #72287280, #72287302, … |
| maximal normal subgroup of | #72287080, #72287280, #72287302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344167);