Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(6,10,11,18,12,19,13,15);(1,2,3,4)(7,14)(8,16)(10,15)(11,13)(17,20)(18,19);(5,6)(7,15)(8,11)(9,12)(10,14)(13,16)(17,18)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12310, #72216412, #72216414, … |
| maximal normal subgroup of | C4×#12310, #72216412, #72216414, … |
| maximal quotient of | #72216706, #72216708, #72216709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44061);