Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
396 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 100 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)(5,6,9,11,8,7,10,12)(13,14,15,22,18,21,17,27,19,23,16,25,24,28,26,20);(1,2,3,4)(5,7,9,12,8,6,10,11)(14,20,23,27)(15,17)(16,26)(18,24)(21,25,28,22);(6,7)(11,12)(13,15)(14,21)(16,19)(17,24)(18,26)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285812, #72285843, #72285857, … |
| maximal normal subgroup of | #72285812, #72285843, #72285857, … |
| maximal quotient of | #72287268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43935);