Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14516 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
879 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)(4,20,25,9)(5,26,30,41)(6,31)(7,12)(8,17)(10,28,42,29)(11,14)(15,36,45,49)(16,33,46,56)(18,34)(19,35)(21,32,48,59)(22,52)(23,38,24,39)(27,43,47,40)(37,55)(44,63)(50,53)(51,60)(54,61)(57,62)(58,64);(1,3,14,17,7,19,6,18)(2,8,11,35,12,34,31,13)(4,21,23,15,25,48,24,45)(5,27,28,16,30,47,29,46)(9,36,39,59,20,49,38,32)(10,40,26,56,42,43,41,33)(22,44,55,57,54,64,53,51)(37,58,61,60,50,63,52,62);(1,4,7,25)(2,9)(3,15,19,45)(5,22,30,54)(6,23,14,24)(8,32)(10,37)(11,38)(12,20)(13,36)(16,44,46,64)(17,21,18,48)(26,52)(27,57,47,51)(28,53,29,55)(31,39)(33,58)(34,59)(35,49)(40,62)(41,61)(42,50)(43,60)(56,63);(1,5)(2,10,12,42)(3,16)(4,22)(6,29)(7,30)(8,33,34,56)(9,37,20,50)(11,26,31,41)(13,43,35,40)(14,28)(15,44)(17,47)(18,27)(19,46)(21,51)(23,55)(24,53)(25,54)(32,58,59,63)(36,60,49,62)(38,52,39,61)(45,64)(48,57)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=efgi,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295320);