Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10254 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1444 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,6,4)(9,10,13,18,15,19,14,17)(11,20,21,12,16,23,22,24);(3,6)(4,8)(9,11,13,21,15,16,14,22)(10,12,17,20,19,24,18,23);(1,3,4,2,5,6,8,7)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,21)(18,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215706, #72216075, #72265410, … |
| maximal normal subgroup of | #72215706, #72216075, #72265410, … |
| maximal quotient of | #72215669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28779);