Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14616 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,19,34,18,41,17,9)(4,20,25,50,24,10,23,36)(5,26,30,11,29,38,28,39)(15,40,44,48,21,51,43,32)(16,45,47,62,46,33,27,37)(22,53,56,35,57,61,55,49)(42,59,54,60,52,63,58,64);(1,3)(2,9,13,34)(4,21,24,15)(5,27,29,47)(6,17)(7,19)(8,18)(10,32)(11,37)(12,14,31,41)(16,28,46,30)(20,48)(22,54)(23,44,25,43)(26,33)(35,60,49,64)(36,40)(38,45)(39,62)(42,55)(50,51)(52,56)(53,59,61,63)(57,58);(1,4)(2,10,13,20)(3,15)(5,22)(6,23)(7,25)(8,24)(9,32,34,48)(11,35,39,49)(12,36,31,50)(14,40,41,51)(16,42)(17,43)(18,21)(19,44)(26,53,38,61)(27,58)(28,55)(29,57)(30,56)(33,59,45,63)(37,60,62,64)(46,52)(47,54);(1,5,6,28,8,29,7,30)(2,11,31,26,13,39,12,38)(3,16,19,47,18,46,17,27)(4,22,23,55,24,57,25,56)(9,33,14,37,34,45,41,62)(10,35,50,53,20,49,36,61)(15,42,44,54,21,52,43,58)(32,59,40,60,48,63,51,64)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eh,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298346);