Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149797 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#3220 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,4)(5,6)(7,16,18,11)(8,12)(9,14,24,13)(10,15)(17,31,32,25)(19,26,23,27)(20,33,34,36)(21,29)(22,28,35,30);(5,7)(6,11)(8,20)(9,23)(10,19)(12,25)(13,26)(14,16)(15,27)(17,21)(18,24)(22,32)(28,31)(29,36)(30,33)(34,35);(2,4)(5,8)(6,12)(7,17)(9,22)(10,21)(11,25)(13,30)(14,28)(15,29)(16,31)(18,32)(19,20)(23,34)(24,35)(26,33)(27,36)Polycyclic presentation:
2^9|a^2=f,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,f^b=fgh,g^b=gi,d^c=di,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59307);