Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #493 |
| 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×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,16)(12,14)(15,19)(18,20)(22,26)(23,27)(24,30)(28,32)(29,36)(31,35)(33,34);(2,4)(6,10)(11,18)(12,17)(13,19)(21,22,27,34,24,29,31,32,25,28,35,36,30,33,23,26);(5,7,14,20,8,15,9,16)(6,11,17,19,10,18,12,13)(22,28)(26,32)(29,33)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=eh,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=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48887);