Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14043 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1221 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)(3,13,17,8)(4,19,22,9)(5,24,26,10)(6,11)(7,27)(12,14)(15,29)(16,30)(18,31,28,36)(20,45)(21,46)(23,34,39,47)(25,35,40,52)(32,49)(33,50)(37,42)(38,44)(41,54,61,59)(43,58)(48,62,64,56)(51,57)(53,55)(60,63);(1,3,14,18,6,17,7,28)(2,8,12,36,11,13,27,31)(4,20,39,42,22,49,23,15)(5,16,40,43,26,44,25,53)(9,32,34,29,19,45,47,37)(10,30,35,58,24,38,52,55)(21,48,60,61,50,64,51,41)(33,56,57,54,46,62,63,59);(1,4,7,23,6,22,14,39)(2,9,12,34,11,19,27,47)(3,15,18,20,17,42,28,49)(5,21,25,51,26,50,40,60)(8,29,31,32,13,37,36,45)(10,33,35,57,24,46,52,63)(16,41,43,48,44,61,53,64)(30,54,55,56,38,59,58,62);(1,5)(2,10)(3,16)(4,21)(6,26)(7,25)(8,30)(9,33)(11,24)(12,35)(13,38)(14,40)(15,41)(17,44)(18,43)(19,46)(20,48)(22,50)(23,51)(27,52)(28,53)(29,54)(31,55)(32,56)(34,57)(36,58)(37,59)(39,60)(42,61)(45,62)(47,63)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277312);