Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 36 points:
(5,6,8,15,10,16,24,32,11,14,22,30,9,13,23,29)(7,17,19,31,21,28,34,36,18,12,25,27,20,26,33,35);(1,2,3,4)(5,7,9,20,11,18,10,21)(6,12,13,28,14,17,16,26)(8,19,23,33,22,25,24,34)(15,31,29,35,30,27,32,36);(6,13,14,16)(7,18)(8,22)(12,26,17,28)(15,32,30,29)(20,21)(23,24)(27,36,31,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239962, #72250581, #72250952, … |
| maximal normal subgroup of | #72239962, #72250581, #72250952, … |
| maximal quotient of | #72240005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34182);