Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124660 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1510 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7607 |
| prime radical | C24 | #104 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,11,13,24,14,28,19,31)(4,15,17,8)(5,10)(6,9)(7,12,21,30,22,29,23,16)(18,27,32,25)(20,26);(1,3)(2,7,9,22)(4,16)(5,19)(6,14)(8,24,15,31)(10,21,26,23)(11,27,28,25)(12,18)(13,20)(17,30)(29,32);(4,17)(8,15)(11,28)(12,29)(16,30)(18,32)(24,31)(25,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219607, #72226636, #72226637 |
| maximal normal subgroup of | #72219607, #72226636, #72226637 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31872);