Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
827 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 827 conjugacy classes of subgroups — too many to draw. Only its 182 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:
(1,2,4,7,5,8,3,6)(9,10,14,17,13,18,12,15)(11,19,22,23,20,16,21,24);(10,15,18,17)(11,20)(16,24,19,23)(21,22);(2,6,8,7)(9,11,12,21,13,20,14,22)(10,16,15,23,18,19,17,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72213016, C2×#110276, #72215973, … |
| maximal normal subgroup of | #72213016, C2×#110276, #72215973, … |
| maximal quotient of | #72212999, C2×#110276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17472);