Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,18)(19,27)(20,31)(21,26)(22,24)(23,28)(25,33)(29,45)(30,41)(32,40)(34,39)(35,38)(36,44)(37,46)(42,47)(43,48);(2,7,10,9)(8,14,16,15)(17,19,21,29,22,30,35,47)(18,23,26,37,24,36,38,48)(20,28,32,43,33,44,34,46)(25,27,39,42,31,41,40,45);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(18,24)(23,36)(25,31)(26,38)(27,41)(37,48)(39,40)(42,45)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36165);