Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422310 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15304 |
| 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 | |
1129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,30)(3,14,17,35,18,34,29,9)(4,19,23,38,24,37,22,10)(5,25,27,11,28,40,31,41)(15,32,45,60,20,50,44,42)(16,46,26,57,47,33,48,39)(21,52,55,62,56,36,54,49)(43,61,53,59,51,64,58,63);(1,3,8,18)(2,9,13,35)(4,20)(5,26,28,48)(6,29,7,17)(10,32)(11,39,41,57)(12,34,30,14)(15,24)(16,31,47,27)(19,42)(21,53)(22,44)(23,45)(25,46,40,33)(36,61)(37,60)(38,50)(43,55)(49,63)(51,54)(52,64)(56,58)(59,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,32)(11,36)(12,37)(13,38)(14,42)(16,43)(17,44)(18,20)(19,30)(25,49)(26,58)(27,54)(28,56)(29,45)(31,55)(33,59)(34,60)(35,50)(39,61)(40,62)(41,52)(46,63)(47,51)(48,53)(57,64);(1,5,6,27,8,28,7,31)(2,11,12,40,13,41,30,25)(3,16,17,26,18,47,29,48)(4,21,22,54,24,56,23,55)(9,33,14,39,35,46,34,57)(10,36,37,62,38,52,19,49)(15,43,44,58,20,51,45,53)(32,59,42,61,50,63,60,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329506);