Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1289 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,21)(12,26,28,52)(13,31)(14,19)(15,20)(16,22,34,46)(17,30,37,54)(18,39)(23,49)(24,43)(25,44)(27,40)(29,56,55,38)(32,41)(33,59)(35,62,60,70)(36,48,61,67)(42,63)(45,65)(47,69,68,58)(50,66)(51,71)(53,57,72,64);(2,6,8,7)(9,11,13,23,14,24,32,50,15,25,33,51,27,45,39,63)(10,16,18,35,40,61,59,69,20,34,41,60,19,36,31,58)(12,22,29,47,30,48,57,70,28,46,55,68,54,67,64,62)(17,21,38,42,52,65,72,71,37,44,56,66,26,43,53,49)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fgh,f^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 554);