Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135526 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8747 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 60 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)(4,15,17,8)(5,18,20,9)(6,10)(7,13,22,30)(12,25,28,32)(14,26)(16,23)(19,24)(21,27)(29,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,11)(10,22)(15,31)(16,27)(17,28)(18,26)(19,29)(20,30)(23,32)(24,25);(1,4,5,19)(2,8,9,24)(3,12,13,29)(6,17,20,16)(7,21,11,25)(10,15,18,23)(14,28,30,27)(22,31,26,32)Polycyclic presentation:
2^9|a^2=f,c^2=dh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=eg,f^c=fh,h^c=hi,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72314472, #72314493, #72338693, … |
| maximal normal subgroup of | #72314472, #72314493, #72338693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42722);