Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10876 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,11,26,36,8,13,27,37,5,23,25,50)(3,14,18,40,19,30,47,56,17,39,46,62,16,9,28,31)(4,10,21,33,20,35,48,59,22,34,49,58,24,32,51,57)(15,41,44,63,42,29,53,55,43,38,60,61,45,54,52,64);(1,3)(2,9,13,30)(4,15)(5,16)(6,18)(7,19)(8,17)(10,29,34,54)(11,14,23,39)(12,31,37,56)(20,42)(21,44)(22,43)(24,45)(25,28)(26,47)(27,46)(32,41,35,38)(33,55,58,64)(36,40,50,62)(48,53)(49,60)(51,52)(57,63,59,61);(1,4,6,21,7,20,26,48,8,22,27,49,5,24,25,51)(2,10,12,33,11,35,36,59,13,34,37,58,23,32,50,57)(3,15,18,44,19,42,47,53,17,43,46,60,16,45,28,52)(9,29,31,55,14,38,40,61,30,54,56,64,39,41,62,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239196, #72239197, #72239200, … |
| maximal normal subgroup of | #72239196, #72239197, #72239200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54345);