Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | #532 | C23 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 621 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)(3,12)(4,17)(5,10)(6,11)(7,14)(8,18)(9,21)(13,28)(15,29)(16,25)(19,24)(20,26)(22,31)(23,30)(27,32);(4,18)(5,21)(7,22)(8,17)(12,28)(14,19)(23,24)(25,29)(26,27)(30,31);(1,3)(2,7)(4,19)(5,13)(6,16)(8,25)(9,22)(10,23)(11,24)(12,17)(14,18)(15,21)(20,31)(26,28)(27,29)(30,32)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,e^b=ei,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72317815, #72338410, #72338411 |
| maximal normal subgroup of | #72317815, #72338410, #72338411 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60317);