Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14)(15,25,27,34)(16,22)(17,31)(18,24)(19,21)(20,26,35,29)(23,32)(28,42,41,39)(30,38)(33,37)(36,43,44,40);(2,4)(13,15,17,28,18,29,33,43,19,27,32,41,16,26,30,40)(14,20,23,36,24,34,38,42,21,35,31,44,22,25,37,39);(5,7,8,12,9,6,10,11)(14,21)(15,26,27,29)(17,32)(20,34,35,25)(22,24)(28,43,41,40)(30,33)(36,39,44,42)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72321078, #72321079, #72321099, … |
| maximal normal subgroup of | #72321078, #72321079, #72321099, … |
| maximal quotient of | #72322400, #72322514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35456);