Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17145 |
| 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 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,7,12,6,13)(3,14,17,43,18,9,29,32)(4,19,26,10,22,39,21,34)(5,11,24,38,25,37,27,40)(15,44,46,30,47,51,49,41)(16,31,50,35,48,42,23,57)(20,52,54,33,55,62,56,36)(45,61,58,60,64,63,53,59);(1,3,6,17,7,18,8,29)(2,9,13,32,12,14,28,43)(4,15,21,46,22,47,26,49)(5,23,24,48,25,50,27,16)(10,30,19,51,34,41,39,44)(11,35,38,31,37,57,40,42)(20,53,54,64,55,58,56,45)(33,61,62,59,36,63,52,60);(1,4,7,22)(2,10,12,34)(3,15,18,47)(5,20,25,55)(6,26,8,21)(9,30,14,41)(11,36,37,33)(13,39,28,19)(16,45,48,64)(17,49,29,46)(23,58,50,53)(24,56,27,54)(31,60,42,59)(32,44,43,51)(35,61,57,63)(38,62,40,52);(1,5)(2,11)(3,16)(4,20)(6,27)(7,25)(8,24)(9,31)(10,33)(12,37)(13,40)(14,42)(15,45)(17,50)(18,48)(19,52)(21,56)(22,55)(23,29)(26,54)(28,38)(30,59)(32,35)(34,36)(39,62)(41,60)(43,57)(44,63)(46,58)(47,64)(49,53)(51,61)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385584);