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)(9,10,16,23,14,21,17,42)(11,24,29,52,27,18,30,45)(12,31,37,61,35,47,38,67)(13,20,40,51,33,22,15,41)(19,34,49,66,48,65,60,36)(25,53,56,71,32,62,57,72)(26,50,28,59,39,44,58,68)(43,54,69,55,46,63,70,64);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,18,21,24)(12,32,35,25)(13,39,33,26)(15,28,40,58)(16,29,17,30)(19,46,48,43)(20,50,22,44)(23,45,42,52)(31,53,47,62)(34,54,65,63)(36,64,66,55)(37,57,38,56)(41,68,51,59)(49,70,60,69)(61,71,67,72);(9,12,33,65,17,38,40,66,14,35,13,34,16,37,15,36)(10,19,41,61,23,49,20,47,21,48,51,67,42,60,22,31)(11,25,26,54,30,57,58,64,27,32,39,63,29,56,28,55)(18,43,68,72,45,69,50,53,24,46,59,71,52,70,44,62);(1,4,5,3)(2,6,7,8)(9,13,17,15,14,33,16,40)(10,20,42,41,21,22,23,51)(11,26,30,58,27,39,29,28)(12,34,38,36,35,65,37,66)(18,44,52,59,24,50,45,68)(19,47,60,61,48,31,49,67)(25,54,57,64,32,63,56,55)(43,62,70,71,46,53,69,72)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,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403360);