Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17086 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,15,18,34,20,9,19,33)(4,21,23,37,24,42,30,10)(5,25,27,11,28,41,29,40)(16,44,46,31,47,52,51,35)(17,48,26,43,49,38,50,32)(22,36,55,62,56,39,57,53)(45,60,54,63,64,59,58,61);(1,3,8,20)(2,9,12,15)(4,16,24,47)(5,26,28,50)(6,19,7,18)(10,35,37,31)(11,38,40,48)(13,34,14,33)(17,29,49,27)(21,52,42,44)(22,54,56,58)(23,51,30,46)(25,32,41,43)(36,61,39,63)(45,57,64,55)(53,60,62,59);(1,4)(2,10)(3,16)(5,22)(6,30)(7,23)(8,24)(9,31)(11,39)(12,37)(13,42)(14,21)(15,35)(17,45)(18,51)(19,46)(20,47)(25,53)(26,58)(27,57)(28,56)(29,55)(32,60)(33,44)(34,52)(36,40)(38,63)(41,62)(43,59)(48,61)(49,64)(50,54);(1,5)(2,11)(3,17)(4,22)(6,29)(7,27)(8,28)(9,32)(10,36)(12,40)(13,25)(14,41)(15,43)(16,45)(18,50)(19,26)(20,49)(21,53)(23,57)(24,56)(30,55)(31,59)(33,38)(34,48)(35,60)(37,39)(42,62)(44,61)(46,58)(47,64)(51,54)(52,63)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383486);