Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #379 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
753 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 753 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,8)(5,6)(10,14,15,16)(18,21,22,23);(1,3)(2,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16);(1,4)(2,3)(5,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=ei,f^c=fgi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72227833, #72227845, #72227847, … |
| maximal normal subgroup of | #72227833, #72227845, #72227847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4721);