Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16512 |
| 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 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,26,37)(5,11,30,42)(6,13,7,12)(9,19,34,18)(10,24,38,25)(16,44,47,31)(17,48,52,43)(22,35,46,59)(23,53,54,39)(27,56,55,36)(28,41,29,40)(32,51,61,50)(45,62,57,58)(49,60,63,64);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,25,47,26,46,24,16)(5,17,28,50,30,52,29,51)(10,35,21,44,38,59,37,31)(11,32,40,48,42,61,41,43)(23,49,27,57,54,63,55,45)(36,60,39,62,56,64,53,58);(1,4,6,25,8,26,7,24)(2,10,12,21,14,38,13,37)(3,16,18,22,20,47,19,46)(5,27,29,54,30,55,28,23)(9,31,15,35,34,44,33,59)(11,39,41,56,42,53,40,36)(17,49,51,57,52,63,50,45)(32,60,43,62,61,64,48,58);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,41)(13,40)(14,42)(15,43)(16,45)(18,51)(19,50)(20,52)(21,39)(22,49)(24,55)(25,27)(26,54)(31,58)(33,48)(34,61)(35,60)(37,53)(38,56)(44,62)(46,63)(47,57)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372165);