Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477203 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17105 |
| 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 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,28,8,12,7,13)(3,9,29,33,16,35,17,34)(4,18,21,10,23,51,22,38)(5,24,26,41,27,11,30,42)(14,43,53,59,45,36,19,31)(15,46,25,57,48,32,49,39)(20,50,54,37,55,61,56,40)(44,62,64,63,47,60,52,58);(1,3,7,29,8,16,6,17)(2,9,28,34,12,35,13,33)(4,19,22,14,23,53,21,45)(5,25,30,48,27,49,26,15)(10,36,51,59,38,43,18,31)(11,39,42,32,24,57,41,46)(20,44,56,64,55,47,54,52)(37,58,61,60,40,63,50,62);(1,4)(2,10,12,38)(3,14,16,45)(5,20)(6,21)(7,22)(8,23)(9,31)(11,40,24,37)(13,18,28,51)(15,47,48,44)(17,19,29,53)(25,52,49,64)(26,54)(27,55)(30,56)(32,58)(33,43)(34,36)(35,59)(39,62)(41,61,42,50)(46,63)(57,60);(1,5)(2,11,12,24)(3,15,16,48)(4,20)(6,26)(7,30)(8,27)(9,32)(10,37,38,40)(13,41,28,42)(14,44,45,47)(17,49,29,25)(18,50,51,61)(19,52,53,64)(21,54)(22,56)(23,55)(31,58)(33,39)(34,57)(35,46)(36,60)(43,62)(59,63)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,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, 384400);