Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2333 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14481 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
755 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)(3,13,17,37)(4,9)(5,21,24,50)(6,11)(7,12)(8,16,29,25)(10,23,36,52)(14,39,42,38)(15,28)(18,44,47,62)(19,32)(20,33)(22,34)(26,35)(27,48,30,41)(31,46,58,64)(40,55)(43,54)(45,56)(49,57)(51,60)(53,61)(59,63);(1,3)(2,8)(4,14)(5,22)(6,25)(7,17)(9,30)(10,34)(11,37)(12,29)(13,35)(15,23)(16,26)(18,45)(19,48)(20,42)(21,28)(24,51)(27,33)(31,56)(32,38)(36,60)(39,57)(40,46)(41,49)(43,52)(44,55)(47,63)(50,54)(53,62)(58,59)(61,64);(1,4,6,19,7,20,26,49)(2,9,11,32,12,33,35,57)(3,14,25,48,17,42,16,41)(5,18,23,46,24,47,52,64)(8,27,37,39,29,30,13,38)(10,31,21,44,36,58,50,62)(15,40,51,63,43,61,22,45)(28,53,60,56,54,55,34,59);(1,5,6,23,7,24,26,52)(2,10,35,50,12,36,11,21)(3,15,16,22,17,43,25,51)(4,18,19,46,20,47,49,64)(8,28,37,60,29,54,13,34)(9,31,57,62,33,58,32,44)(14,40,41,45,42,61,48,63)(27,53,39,56,30,55,38,59)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294769);