Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14576 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
797 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)(3,13,17,38)(4,18)(5,23,26,40)(6,11)(7,12)(8,16,33,27)(9,22)(10,25,37,29)(14,44,19,42)(15,32)(20,53,56,58)(21,39)(24,36)(28,35)(30,41)(31,46,50,48)(34,55,51,57)(43,49)(45,63)(47,60)(52,59)(54,61)(62,64);(1,3)(2,8)(4,19,22,14)(5,24)(6,27)(7,17)(9,31,18,50)(10,36)(11,38)(12,33)(13,41)(15,25)(16,30)(20,54,56,64)(21,46,28,48)(23,32)(26,49)(29,47)(34,61,51,62)(35,42,39,44)(37,43)(40,60)(45,55,52,57)(53,59,58,63);(1,4)(2,9,12,18)(3,14)(5,20)(6,28)(7,22)(8,31,33,50)(10,34,37,51)(11,39,41,35)(13,42,38,44)(15,45)(16,48)(17,19)(21,30)(23,58,40,53)(24,54)(25,57)(26,56)(27,46)(29,55)(32,59,60,63)(36,62,43,61)(47,52)(49,64);(1,5)(2,10)(3,15)(4,20)(6,29)(7,26)(8,32)(9,34)(11,40)(12,37)(13,43)(14,45)(16,49)(17,47)(18,51)(19,52)(21,57)(22,56)(23,41)(24,27)(25,30)(28,55)(31,59)(33,60)(35,58)(36,38)(39,53)(42,61)(44,62)(46,54)(48,64)(50,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,e^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297785);