Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,10)(9,17)(11,15)(12,16)(14,20)(18,19)(22,26)(23,27)(24,30)(28,32)(29,36)(31,35)(33,34);(6,10)(11,19)(12,17)(13,20)(21,22,27,34,24,29,31,32,25,28,35,36,30,33,23,26);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)(22,28)(26,32)(29,33)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,c^2=h,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42207);