Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1555 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7592 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,12,14,23,15,22,19,7)(4,16,17,8)(5,9)(6,11)(10,20)(13,27,28,31,30,26,29,21)(18,25,32,24);(1,3)(2,7)(4,13)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(14,20)(16,31)(17,30)(18,28)(24,27)(25,26)(29,32);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,24)(10,25)(11,16)(12,26)(14,29)(15,30)(19,28)(20,32)(22,27)(23,31)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338318, #72338325, #72338394, … |
| maximal normal subgroup of | #72338318, #72338325, #72338394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31716);