Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×2−1+4 | D4 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 559 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2)(3,4)(5,6,11,15)(7,12,10,26)(8,13,21,32)(9,22,25,28)(14,23,34,17)(16,27,19,31)(18,30,35,33)(20,36,24,29);(5,7,11,10)(6,12,15,26)(8,18,21,35)(9,23,25,17)(13,29,32,36)(14,22,34,28)(16,24,19,20)(27,33,31,30);(2,4)(5,8)(6,13)(7,16)(9,24)(10,19)(11,21)(12,27)(14,33)(15,32)(17,35)(18,23)(20,25)(22,29)(26,31)(28,36)(30,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60338);