Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10,14,21,17,24,16,23)(11,25,28,46,30,45,29,18)(12,31,34,51,37,50,36,19)(13,38,15,41,40,20,39,22)(26,55,57,53,59,43,58,71)(27,52,61,44,62,54,42,69)(32,47,65,72,56,70,60,63)(33,66,35,48,68,49,67,64);(1,3,5,8)(2,6,4,7)(9,11)(10,18)(12,32)(13,27)(14,29)(15,42)(16,28)(17,30)(19,47)(20,44)(21,45)(22,52)(23,25)(24,46)(26,35)(31,63)(33,57)(34,60)(36,65)(37,56)(38,69)(39,61)(40,62)(41,54)(43,49)(48,71)(50,72)(51,70)(53,64)(55,66)(58,68)(59,67);(2,4)(6,7)(9,12,17,37)(10,19,24,51)(11,26,30,59)(13,35,40,67)(14,36,16,34)(15,33,39,68)(18,43,46,55)(20,49,38,66)(21,50,23,31)(22,48,41,64)(25,53,45,71)(27,60,62,65)(28,58,29,57)(32,61,56,42)(44,63,69,72)(47,54,70,52);(1,4,5,2)(3,7,8,6)(9,13,14,15,17,40,16,39)(10,20,21,22,24,38,23,41)(11,27,29,42,30,62,28,61)(12,33,34,35,37,68,36,67)(18,44,45,52,46,69,25,54)(19,48,31,49,51,64,50,66)(26,56,58,65,59,32,57,60)(43,70,53,72,55,47,71,63)Polycyclic presentation:
2^9|a^2=e,b^2=eh,c^2=i,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415911);