Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431743 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#127 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#399 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3078 conjugacy classes of subgroups; only the 904 normal subgroups are shown.
This group has 3078 conjugacy classes of subgroups — too many to draw. Only its 904 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:
(1,2,5,10,6,8,12,19)(3,11,14,21,15,24,18,7)(4,9,17,23,16,22,26,30)(13,25,28,31,27,32,29,20);(1,3,12,14,6,15,5,18)(2,7,19,11,8,21,10,24)(4,13,26,28,16,27,17,29)(9,20,30,25,22,31,23,32);(2,8)(9,22)(10,19)(23,30);(1,4)(2,9)(3,13)(5,17)(6,16)(7,20)(8,22)(10,23)(11,25)(12,26)(14,28)(15,27)(18,29)(19,30)(21,31)(24,32);(4,16)(9,22)(13,27)(17,26)(20,31)(23,30)(25,32)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=fg,f^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286055, #72286143, #72286225, … |
| maximal normal subgroup of | #72286055, #72286143, #72286225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339595);