Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,15)(7,17,20,35)(8,13,23,14)(10,16,25,29)(11,28,31,18)(12,22,33,19)(21,32,26,30)(24,34,27,36);(1,3,4,2)(5,7,9,20)(6,11,15,31)(8,18,23,28)(10,21,25,26)(12,29,33,16)(13,17,14,35)(19,24,22,27)(30,36,32,34);(5,7,9,20)(6,12,15,33)(8,22,23,19)(10,26,25,21)(11,29,31,16)(13,17,14,35)(18,27,28,24)(30,34,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gh,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59656);