Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16266 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 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)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,44)(17,48)(19,34)(20,33)(22,31)(23,54)(24,37)(26,36)(27,57)(28,41)(29,40)(30,39)(32,50)(35,56)(38,55)(42,47)(43,51)(45,63)(46,60)(49,61)(52,62)(53,59)(58,64);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,25,46,26,16,24,47)(5,17,28,50,29,51,30,52)(10,31,21,42,37,44,36,60)(11,32,39,48,40,62,41,43)(23,53,27,58,55,45,56,49)(35,59,38,61,57,63,54,64);(1,4)(2,10)(3,16)(5,27)(6,25)(7,24)(8,26)(9,31)(11,38)(12,21)(13,36)(14,37)(15,42)(17,49)(18,47)(19,46)(20,22)(23,28)(29,56)(30,55)(32,61)(33,60)(34,44)(35,39)(40,54)(41,57)(43,63)(45,50)(48,59)(51,58)(52,53)(62,64);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,35)(12,41)(13,39)(14,40)(15,43)(16,45)(18,52)(19,50)(20,51)(21,38)(22,53)(24,56)(25,27)(26,55)(31,59)(33,48)(34,62)(36,54)(37,57)(42,61)(44,63)(46,58)(47,49)(60,64)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362772);