Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461099 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16379 |
| 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 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,12,6,13)(3,15,18,33,19,9,29,32)(4,20,28,10,23,36,22,35)(5,24,27,11,26,39,25,38)(16,42,44,30,45,51,50,40)(17,46,47,31,49,56,48,41)(21,37,55,52,54,34,53,59)(43,57,61,64,63,58,62,60);(1,3,6,18,7,19,8,29)(2,9,13,32,12,15,14,33)(4,16,22,44,23,45,28,50)(5,17,25,47,26,49,27,48)(10,30,20,51,35,40,36,42)(11,31,24,56,38,41,39,46)(21,43,53,61,54,63,55,62)(34,57,52,64,37,58,59,60);(1,4,7,23)(2,10,12,35)(3,16,19,45)(5,21,26,54)(6,28,8,22)(9,30,15,40)(11,37,38,34)(13,36,14,20)(17,43,49,63)(18,50,29,44)(24,52,39,59)(25,55,27,53)(31,58,41,57)(32,42,33,51)(46,60,56,64)(47,62,48,61);(1,5)(2,11)(3,17)(4,21)(6,25)(7,26)(8,27)(9,31)(10,34)(12,38)(13,24)(14,39)(15,41)(16,43)(18,47)(19,49)(20,52)(22,53)(23,54)(28,55)(29,48)(30,57)(32,56)(33,46)(35,37)(36,59)(40,58)(42,60)(44,61)(45,63)(50,62)(51,64)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,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, 368296);