Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C23 | #444 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(6,11,13,19,14,25,23,35)(7,15,17,32)(8,18)(12,26,28,24,30,34,29,27)(16,33,20,31)(22,36);(1,2,3,4)(5,6)(7,12)(8,19)(9,23)(10,14)(11,22)(13,21)(15,28)(16,34)(17,30)(18,35)(20,26)(24,33)(25,36)(27,31)(29,32);(5,7,8,20,21,32,36,31,10,17,18,16,9,15,22,33)(6,12,19,26,13,29,25,27,14,30,35,34,23,28,11,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216223, #72216321, #72216330, … |
| maximal normal subgroup of | #72216223, #72216321, #72216330, … |
| maximal quotient of | #72223649, C2×#129164, #72242113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36360);