Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153129 |
| 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×#124 | C23 |
| prime radical | #30493 | C2 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 569 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)(8,20)(9,25)(12,30)(13,26)(15,33)(16,18)(17,35)(21,32)(22,36)(23,24)(27,28)(31,34);(5,6,10,14)(7,15,19,31)(8,21,23,12)(9,26,25,13)(11,22,29,17)(16,34,18,33)(20,30,24,32)(27,35,28,36);(2,4)(5,7)(6,11)(8,22)(9,16)(10,19)(12,31)(13,27)(14,29)(15,21)(17,23)(18,25)(20,36)(24,35)(26,28)(30,34)(32,33)Polycyclic presentation:
2^9|b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60360);