Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16373 |
| 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 | |
998 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)(3,13,17,8)(4,18)(5,23)(6,26)(7,11)(9,21)(10,25)(12,28)(14,40,19,48)(15,30,44,38)(16,31,46,39)(20,47)(22,37)(24,36)(27,34)(29,42,32,45)(33,50)(35,53)(41,60,49,64)(43,59,63,57)(51,58)(52,54)(55,61,56,62);(1,3,6,16,7,17,28,46)(2,8,26,39,11,13,12,31)(4,19,21,45,22,14,27,42)(5,15,24,43,25,44,54,63)(9,32,37,48,34,29,18,40)(10,30,52,59,23,38,36,57)(20,49,50,62,51,41,53,61)(33,56,58,64,35,55,47,60);(1,4)(2,9)(3,14)(5,20)(6,27)(7,22)(8,29)(10,35)(11,34)(12,37)(13,32)(15,41)(16,45)(17,19)(18,26)(21,28)(23,33)(24,53)(25,51)(30,56)(31,40)(36,47)(38,55)(39,48)(42,46)(43,62)(44,49)(50,54)(52,58)(57,64)(59,60)(61,63);(1,5,7,25)(2,10)(3,15,17,44)(4,20,22,51)(6,24,28,54)(8,30)(9,33)(11,23)(12,36)(13,38)(14,41,19,49)(16,43,46,63)(18,47)(21,50,27,53)(26,52)(29,55)(31,57)(32,56)(34,35)(37,58)(39,59)(40,60)(42,61,45,62)(48,64)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368056);