Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C2×#135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,23,14,21,17,41)(11,24,29,51,27,18,30,44)(12,31,36,62,34,45,37,66)(13,20,39,49,32,22,15,40)(19,33,47,65,46,63,61,35)(25,52,56,70,54,60,57,72)(26,48,28,59,38,43,58,67)(42,53,68,55,50,64,69,71);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,18,21,24)(12,25,34,54)(13,38,32,26)(15,28,39,58)(16,29,17,30)(19,42,46,50)(20,48,22,43)(23,44,41,51)(31,60,45,52)(33,64,63,53)(35,55,65,71)(36,56,37,57)(40,67,49,59)(47,68,61,69)(62,72,66,70);(9,12,32,63,17,37,39,65,14,34,13,33,16,36,15,35)(10,19,40,62,23,47,20,45,21,46,49,66,41,61,22,31)(11,25,26,53,30,57,58,71,27,54,38,64,29,56,28,55)(18,42,67,72,44,68,48,52,24,50,59,70,51,69,43,60);(1,4,5,3)(2,6,7,8)(9,13,17,15,14,32,16,39)(10,20,41,40,21,22,23,49)(11,26,30,58,27,38,29,28)(12,33,37,35,34,63,36,65)(18,43,51,59,24,48,44,67)(19,45,61,62,46,31,47,66)(25,53,57,71,54,64,56,55)(42,60,69,70,50,52,68,72)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403357);