Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,3)(5,6,8,11,9,7,10,12)(13,14,15,19,18,23,17,21)(16,25,24,28,26,20,27,22);(1,2,4,3)(5,7,8,12,9,6,10,11)(14,19,23,21)(16,26)(20,28,25,22)(24,27);(1,3,4,2)(14,20)(15,17)(16,24)(19,28)(21,22)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216334, #72227144, #72227150, … |
| maximal normal subgroup of | #72216334, #72227144, #72227150, … |
| maximal quotient of | #72216321, #72216338, #72225364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47317);