Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14023 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1648 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)(3,14)(4,19,23,37)(5,24)(6,27)(7,12)(8,13)(9,17)(10,22,38,30)(11,25)(15,35,20,31)(16,32)(18,33)(21,52,55,36)(26,39)(28,34)(29,40)(41,45,49,44)(42,46)(43,61,51,64)(47,57)(48,58)(50,54,60,53)(56,62,59,63);(1,3)(2,9)(4,20,23,15)(5,16)(6,28)(7,17)(8,18)(10,35,38,31)(11,32)(12,33)(13,34)(14,27)(19,49,37,41)(21,51,55,43)(22,44,30,45)(24,42)(25,47)(26,48)(29,46)(36,59,52,56)(39,57)(40,58)(50,64,60,61)(53,62,54,63);(1,4,6,22,8,23,7,30)(2,10,27,19,13,38,12,37)(3,15,17,44,18,20,28,45)(5,21,29,54,26,55,25,53)(9,31,33,49,34,35,14,41)(11,36,24,50,40,52,39,60)(16,43,47,63,48,51,46,62)(32,56,57,64,58,59,42,61);(1,5)(2,11)(3,16)(4,21)(6,29)(7,25)(8,26)(9,32)(10,36)(12,39)(13,40)(14,42)(15,43)(17,47)(18,48)(19,50)(20,51)(22,54)(23,55)(24,27)(28,46)(30,53)(31,56)(33,57)(34,58)(35,59)(37,60)(38,52)(41,61)(44,63)(45,62)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276048);