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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,17,12,15,27,29,13,18,10,26,9,8,20,16)(7,19,23,31,24,33,36,35,25,21,28,34,22,14,30,32);(1,2,4,3)(5,7,20,30,9,22,10,28,13,25,27,36,12,24,11,23)(6,14,29,31,8,21,17,32,18,33,16,34,15,19,26,35);(1,3,4,2)(5,8)(6,12)(7,21)(9,18)(10,16)(11,29)(13,15)(14,24)(17,20)(19,25)(22,33)(23,31)(26,27)(28,34)(30,32)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,h^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 | #72319447, #72319558, #72319561 |
| maximal normal subgroup of | #72319447, #72319558, #72319561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47319);