Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#423 | C22 |
| socle | C22 | #1325 |
| prime radical | C24 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C2×#423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(5,6,8,14,10,15,23,31,11,16,25,32,9,13,24,29)(7,17,18,30,20,12,22,26,21,27,34,35,19,28,33,36);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,15,27,16,28,13,17)(8,22,23,34,25,33,24,18)(14,30,29,36,32,35,31,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,d^b=dfh,d^c=dhi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1305);