Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153143 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,8)(5,17)(6,10)(9,18)(11,26)(12,20)(13,31)(14,24)(15,23)(16,25)(19,28)(21,30)(22,27)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,22)(9,21)(10,20)(14,29)(15,27)(16,28)(17,31)(18,30)(19,23)(24,26)(25,32);(1,4)(2,8)(3,11)(5,14)(6,16)(7,19)(9,23)(10,25)(12,28)(13,27)(15,18)(17,24)(20,26)(21,22)(29,30)(31,32)Polycyclic presentation:
2^9|d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72319456, #72339873, #72339880 |
| maximal normal subgroup of | #72319456, #72339873, #72339880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60339);