Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14427 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 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)(4,18,21,48)(5,23)(6,12)(7,11)(9,22,32,49)(10,24)(14,39)(15,43,45,55)(16,37,26,30)(19,36)(20,31,51,59)(25,35)(27,34)(28,42)(29,44,33,46)(38,53)(40,60)(41,57)(47,52,58,54)(50,56)(61,63)(62,64);(1,3)(2,8)(4,14)(5,15)(6,26)(7,17)(9,28)(10,33)(11,13)(12,37)(16,19)(18,39)(20,50)(21,41)(22,53)(23,55)(24,46)(25,45)(27,44)(29,34)(30,36)(31,56)(32,38)(35,43)(40,51)(42,49)(47,61)(48,57)(52,62)(54,63)(58,64)(59,60);(1,4,19,49,7,21,6,22)(2,9,12,18,11,32,36,48)(3,14,26,53,17,41,16,42)(5,20,24,52,25,51,27,54)(8,28,30,57,13,38,37,39)(10,31,23,47,34,59,35,58)(15,40,44,62,45,50,46,63)(29,56,55,64,33,60,43,61);(1,5)(2,10)(3,15)(4,20)(6,27)(7,25)(8,29)(9,31)(11,34)(12,23)(13,33)(14,40)(16,46)(17,45)(18,47)(19,24)(21,51)(22,54)(26,44)(28,56)(30,55)(32,59)(35,36)(37,43)(38,60)(39,61)(41,50)(42,63)(48,58)(49,52)(53,62)(57,64)Polycyclic presentation:
2^9|a^2=h,c^2=efg,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293567);