Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
776 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,14,18,16,11,15,20)(12,21,22,19,23,24,13,17);(1,3,4,8,5,2,6,7)(9,11)(10,16)(12,19)(13,24)(14,18)(15,20)(17,23)(21,22);(9,12,14,22,16,23,15,13)(10,17,20,24,11,19,18,21);(9,13,15,23,16,22,14,12)(10,17,20,24,11,19,18,21)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216421, #72216426, #72216435, … |
| maximal normal subgroup of | #72216421, #72216426, #72216435, … |
| maximal quotient of | #72216429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138665);