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 | #1466 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 122 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)(5,6,9,15,10,16,25,33,11,12,24,27,8,14,22,31)(7,17,20,34,21,13,28,29,18,30,23,36,19,26,35,32);(6,12)(7,18)(14,16)(19,21)(20,23)(28,35)(29,32)(34,36);(2,4)(5,7,10,21,11,18,8,19)(6,13,14,17,12,26,16,30)(9,23,25,35,24,20,22,28)(15,32,31,36,27,29,33,34)Polycyclic presentation:
2^9|a^2=f,c^2=gh,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 | #72292822, #72292827 |
| maximal normal subgroup of | #72292822, #72292827 |
| maximal quotient of | #72293223, #72293299 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36576);