Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(14,19)(15,24)(16,27)(17,26)(20,23)(21,22)(25,28);(2,4)(6,10)(7,8)(11,12)(13,14)(15,25)(16,20)(17,28)(18,23)(19,24)(21,27)(22,26);(6,11)(7,8)(10,12)(14,20)(15,24)(16,27)(17,26)(19,23)(21,28)(22,25)Polycyclic presentation:
2^9|d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fghi,g^b=gi,d^c=dg,e^c=eh,f^c=fghi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51319);