Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93508 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1289 |
| prime radical | C24 | #103 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,4,7,5,8,3,6)(9,10,13,19)(11,23,25,54)(12,29,31,63)(14,21,37,51)(15,20,36,52)(16,22,38,53)(17,39,40,59)(18,44,46,33)(24,43,57,49)(26,45,61,71)(27,55,60,72)(28,56,62,68)(30,41,64,70)(32,48,65,50)(34,47,35,66)(42,67,69,58);(2,6,8,7)(9,11,14,26,16,28,15,27)(10,17,20,41,22,43,21,42)(12,30,32,24,34,58,33,59)(13,25,36,60,38,62,37,61)(18,45,47,56,48,55,29,23)(19,49,51,70,53,40,52,69)(31,64,44,39,35,67,65,57)(46,72,63,68,50,71,66,54)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,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, 772);