Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143951 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1667 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,6,10,15)(7,16,22,36)(8,12,25,33)(9,26,24,14)(11,27,31,18)(13,34,32,23)(17,30,19,29)(20,28,21,35);(2,4)(5,7)(6,11)(8,17)(9,21)(10,22)(12,16)(13,35)(14,30)(15,31)(18,23)(19,25)(20,24)(26,29)(27,34)(28,32)(33,36);(6,12)(7,17)(8,23)(9,24)(13,14)(15,33)(16,35)(18,20)(19,22)(21,27)(25,34)(26,32)(28,36)(29,30)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51148);