Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,25)(20,24)(21,35)(22,37)(23,28)(26,31)(27,32)(29,38)(30,44)(33,41)(34,40)(36,48)(39,45)(42,46)(43,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,24)(19,22)(20,31)(21,32)(25,41)(26,27)(28,40)(29,45)(30,36)(33,46)(34,47)(35,39)(37,38)(42,48)(43,44);(3,9)(6,12)(10,15)(13,16)(17,19,23,22)(18,25,28,37)(20,29,34,33)(21,30,36,32)(24,38,40,41)(26,42,43,39)(27,35,44,48)(31,46,47,45)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgi,d^b=df,f^b=fh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57317);