Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499261 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3184 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23573 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 32 points:
(1,2)(3,11)(4,16)(5,21)(6,9)(7,14)(8,20)(10,19)(12,25)(13,27)(15,30)(17,23)(18,31)(22,29)(24,28)(26,32);(7,22)(11,27)(23,24)(25,30);(1,3)(2,7)(4,17)(5,12)(6,13)(8,25)(9,22)(10,23)(11,16)(14,20)(15,19)(18,28)(21,24)(26,30)(27,31)(29,32);(3,12,13,15)(4,18)(7,23,22,24)(8,26)(11,25,27,30)(14,17,29,28)(16,31)(20,32)Polycyclic presentation:
2^9|d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72219593, #72226635, … |
| maximal normal subgroup of | #72213937, #72219593, #72226635, … |
| maximal quotient of | #72231508, C2×#499261, #72237285, … |
| automorphism group of | #1369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406457);