Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | #127 | C42 |
| Frattini, cosocle | C2×#492 | C22 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 2 generators on 20 points:
(5,6,7,11,9,13,16,20)(8,10,14,18,17,12,15,19);(1,2,3,4)(6,10)(12,13)(18,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=dfhi,f^b=fhi,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,f^d=fhi,g^d=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72279151, #72309212, #72309216, … |
| maximal normal subgroup of | #72279151, #72309212, #72309216, … |
| maximal quotient of | #72279908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1977);