Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16558 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,6,12,7,26)(3,14,17,32,18,9,28,42)(4,10,27,36,21,35,22,39)(5,23,29,52,24,11,25,38)(15,30,19,48,44,40,47,33)(16,31,45,62,46,41,54,57)(20,34,53,59,50,37,51,61)(43,55,49,60,63,56,64,58);(1,3,7,17,6,18,8,28)(2,9,26,42,12,14,13,32)(4,19,22,44,21,47,27,15)(5,16,25,45,24,46,29,54)(10,33,39,30,35,48,36,40)(11,31,52,62,23,41,38,57)(20,49,51,63,50,64,53,43)(34,58,61,55,37,60,59,56);(1,4)(2,10)(3,15)(5,20)(6,21)(7,27)(8,22)(9,30)(11,37)(12,35)(13,39)(14,40)(16,43)(17,47)(18,44)(19,28)(23,34)(24,50)(25,53)(26,36)(29,51)(31,56)(32,48)(33,42)(38,59)(41,55)(45,64)(46,63)(49,54)(52,61)(57,58)(60,62);(1,5,6,24)(2,11)(3,16,18,46)(4,20,21,50)(7,25,8,29)(9,31)(10,34)(12,23)(13,38)(14,41)(15,43,44,63)(17,45,28,54)(19,49,47,64)(22,51,27,53)(26,52)(30,55)(32,57)(33,58)(35,37)(36,59)(39,61)(40,56)(42,62)(48,60)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374290);