Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle, prime radical | C22 | #1580 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ #1312 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
139 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,3,10,11,6,9,13,16,14,12,7,15)(17,18,21,27,23,28,22,25)(19,29,31,46,33,47,32,43)(20,34,37,44,36,42,38,26)(24,39,40,30,41,35,45,48);(17,19,23,33)(18,24,28,41)(20,35,36,39)(21,32,22,31)(25,40,27,45)(26,43,44,46)(29,34,47,42)(30,38,48,37);(1,3,4,11,5,9,10,16,6,12,13,15,14,2,7,8)(18,25,28,27)(20,36)(24,40,41,45)(26,34,44,42)(29,43,47,46)(30,48)(35,39)(37,38)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=ef,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cfi,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320209, #72320246, #72320279, … |
| maximal normal subgroup of | #72320209, #72320246, #72320279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30165);