Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C25 | 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 | |
660 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2)(3,4)(5,6,9,16,10,13,24,29,11,14,23,30,8,15,22,31)(7,17,20,32,21,26,34,35,18,12,25,27,19,28,33,36);(5,7)(6,12)(8,19)(9,20)(10,21)(11,18)(13,28)(14,17)(15,26)(16,32)(22,33)(23,25)(24,34)(27,30)(29,35)(31,36);(2,4)(6,13,14,15)(7,18)(9,23)(12,26,17,28)(16,31,30,29)(19,21)(22,24)(27,36,32,35)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fg,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292834, #72292835, #72292838 |
| maximal normal subgroup of | #72292834, #72292835, #72292838 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36499);