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, socle, prime radical | C23 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 242 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) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11,13,16,12,15)(17,18,21,25,22,27,19,23)(20,29,31,32,30,24,28,26);(2,5)(6,8)(9,11,14,16,13,15,12,10)(18,23,27,25)(20,30)(24,26,29,32)(28,31);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(18,24)(19,21)(20,31)(23,26)(25,32)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=dgi,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 | #72343346, #72343360, #72343367, … |
| maximal normal subgroup of | #72343346, #72343360, #72343367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47280);