Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14134 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,17,9,18,34,30,33)(4,19,24,36,23,49,22,10)(5,25,26,11,28,37,27,38)(15,41,43,39,20,31,51,47)(16,32,45,57,46,58,44,40)(21,35,52,48,54,59,53,55)(42,60,63,56,50,64,62,61);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,31)(11,32)(12,34)(13,33)(14,29)(15,23)(19,47)(21,50)(22,43)(24,51)(25,40)(26,45)(27,44)(28,46)(35,56)(36,41)(37,57)(38,58)(39,49)(42,54)(48,64)(52,62)(53,63)(55,60)(59,61);(1,4)(2,10,13,36)(3,15,18,20)(5,21)(6,22)(7,24)(8,23)(9,31)(11,35,38,59)(12,19,29,49)(14,39)(16,42,46,50)(17,43,30,51)(25,55,37,48)(26,52)(27,53)(28,54)(32,56)(33,41)(34,47)(40,60)(44,62,45,63)(57,64)(58,61);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,32)(10,35)(12,37)(13,38)(14,40)(15,42)(17,45)(18,46)(19,48)(20,50)(22,53)(23,54)(24,52)(25,29)(30,44)(31,56)(33,58)(34,57)(36,59)(39,60)(41,61)(43,63)(47,64)(49,55)(51,62)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281303);