Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,19,24,22,28,20,23,21,27,30,25,31,32,29,26);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(18,23,27,26)(19,20)(22,31)(24,32,25,28)(29,30);(2,8)(4,5)(7,15)(11,13)(18,24)(19,29)(20,30)(22,31)(23,32)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=eh,f^c=fgh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264918, #72265548, #72265553 |
| maximal normal subgroup of | #72264918, #72265548, #72265553 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46304);