Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32873 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2840 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C2 | C2×#2840 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1992 conjugacy classes of subgroups; only the 488 normal subgroups are shown.
This group has 1992 conjugacy classes of subgroups — too many to draw. Only its 488 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 — 5 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,19,15,21)(13,25,27,20)(16,22)(17,23)(18,24)(26,30,28,31)(29,32);(1,3)(2,7)(4,15)(5,13)(6,14)(8,19)(9,20)(10,11)(12,17)(16,28)(18,27)(21,23)(22,30)(24,25)(26,29)(31,32);(1,4)(2,8)(3,12,14,15)(5,16)(6,17)(7,19,11,21)(9,22)(10,23)(13,26,27,28)(18,29)(20,30,25,31)(24,32);(1,3)(2,7)(4,15)(5,13)(6,14)(8,21)(9,20)(10,11)(12,17)(16,28)(18,27)(19,23)(22,31)(24,25)(26,29)(30,32);(1,5,6,18)(2,9,10,24)(3,13,14,27)(4,16,17,29)(7,20,11,25)(8,22,23,32)(12,26,15,28)(19,30,21,31)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ch,g^b=gi,d^c=dfh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538656);