Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471888 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16808 |
| 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 | |
1105 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)(3,8,16,29)(4,9)(5,21)(6,11)(7,24)(10,22)(12,13)(14,31,41,27)(15,43,45,58)(17,47,39,30)(18,50)(19,33)(20,37)(23,38)(25,34)(26,36)(28,49,56,46)(32,51)(35,53)(40,61,44,64)(42,55,48,57)(52,60)(54,62,59,63);(1,3,13,39,6,16,7,17)(2,8,24,47,11,29,12,30)(4,14,25,48,19,41,20,42)(5,15,22,46,23,45,26,49)(9,31,37,55,33,27,34,57)(10,28,21,43,36,56,38,58)(18,40,51,62,52,44,53,63)(32,59,50,61,35,54,60,64);(1,4,6,19)(2,9,11,33)(3,14,16,41)(5,18,23,52)(7,25,13,20)(8,27,29,31)(10,35,36,32)(12,37,24,34)(15,44,45,40)(17,48,39,42)(21,50,38,60)(22,53,26,51)(28,54,56,59)(30,57,47,55)(43,61,58,64)(46,62,49,63);(1,5)(2,10)(3,15)(4,18)(6,23)(7,26)(8,28)(9,32)(11,36)(12,38)(13,22)(14,40)(16,45)(17,49)(19,52)(20,53)(21,24)(25,51)(27,54)(29,56)(30,58)(31,59)(33,35)(34,60)(37,50)(39,46)(41,44)(42,63)(43,47)(48,62)(55,61)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379085);