Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2250 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,22,20,27,28,25,19,23,16,21,18,15,24,26);(1,2,3,4)(13,15,19,27)(14,20,23,18)(16,22,17,21)(24,26,28,25);(6,9)(7,12)(10,11)(13,14)(15,20)(16,26)(17,25)(18,27)(19,23)(21,24)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gh,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44737);