Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#376 | C22 |
| socle | C22 | #1272 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 400 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 — 2 generators on 36 points:
(1,2,3,4)(5,6,9,14,11,16,10,15)(7,17,19,30,20,34,21,36)(8,13,23,31,24,32,25,33)(12,26,27,35,29,22,28,18);(5,7)(6,12,15,28,16,29,14,27)(8,22,24,26)(9,19)(10,21)(11,20)(13,30,31,34,32,36,33,17)(18,25,35,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93377, #72213050, #72213056, … |
| maximal normal subgroup of | C2×#93377, #72213050, #72213056, … |
| maximal quotient of | #72211432, C2×#93377, #72214266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 573);