Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#97 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 230 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:
(9,10,11,15,12,17,19,23,13,16,18,24,20,14,21,22);(1,2,3,6,4,5,7,8);(2,5)(6,8)(10,14,16,17)(11,18)(15,23,24,22)(19,21)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,c^a=cg,f^a=fi,c^b=ch,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72213975, #72213983, #72214029, … |
| maximal normal subgroup of | #72213975, #72213983, #72214029, … |
| maximal quotient of | #72214029, #72216161, C2×#126544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33740);