Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×#131 | C22 |
| socle | C22 | #1282 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C22×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,20,32,44,14,19,31,43,33,45,58,68)(11,21,23,46,25,47,52,69,24,38,51,61,53,64,71,72)(12,26,27,42,29,54,55,67,28,17,30,39,56,40,57,65)(16,34,35,59,37,60,62,22,36,48,41,50,63,70,66,49);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,19,36)(12,22,28,49)(13,23,31,51)(14,24)(15,25)(17,38)(18,41)(20,37,45,63)(21,26)(27,48)(29,50,56,59)(30,34)(32,52,58,71)(33,53)(35,43)(39,46,42,61)(40,64)(44,66)(47,54)(55,70)(57,60)(62,68)(65,69,67,72)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 683);