Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535045 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59816 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2482 conjugacy classes of subgroups; only the 558 normal subgroups are shown.
This group has 2482 conjugacy classes of subgroups — too many to draw. Only its 558 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)(3,15)(4,20)(5,24)(6,13)(7,11)(8,14)(9,18)(10,22)(12,23)(16,28)(17,26)(19,27)(21,30)(25,29)(31,32);(1,3,6,18,8,19,7,25)(2,9,11,15,14,29,13,27)(4,21,5,17,23,32,22,16)(10,30,12,28,24,31,20,26);(1,4,7,22,8,23,6,5)(2,10,13,20,14,24,11,12)(3,16,18,21,19,17,25,32)(9,26,15,30,29,28,27,31);(2,11,14,13)(9,27,29,15)(10,12,24,20)(26,31,28,30);(1,5,6,23,8,22,7,4)(2,12,11,24,14,20,13,10)(3,17,18,32,19,16,25,21)(9,28,15,31,29,26,27,30)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521281);