Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #322 |
| commutator | C8 | C82 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C22 | #1313 |
| prime radical | C23 | #347 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,8,14,9,12,23,25,10,17,21,30,18,27,34,29,11,16,22,31,24,26,35,36,19,15,32,33,7,13,20,28);(2,4)(6,12,15,13,16,26,17,27)(7,18)(8,21,22,32)(9,24)(14,29,30,36,31,28,33,25)(20,35,34,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216250, #72216317, #72216321, … |
| maximal normal subgroup of | #72216250, #72216317, #72216321, … |
| maximal quotient of | C2×#93997, #72227252, #72286348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1193);