Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1333 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| prime radical | C23 | #360 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,12,13,19,16,29,17,30)(7,18,21,25)(8,22,23,36)(9,26)(10,28)(14,15,33,32,35,27,31,34)(20,24);(2,4)(5,6,8,14,24,34,22,29,28,13,18,31,20,15,7,19,11,16,23,35,26,32,36,12,10,17,25,33,9,27,21,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fghi,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216279, #72216321, #72216330, … |
| maximal normal subgroup of | #72216279, #72216321, #72216330, … |
| maximal quotient of | #72223650, C2×#94057, #72242188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1253);