Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #33021 |
| prime radical | #23994 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 789 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:
(1,2)(3,12)(4,7)(5,10)(6,11)(8,15)(9,17)(13,25)(14,29)(16,21)(18,23)(19,26)(20,30)(22,32)(24,31)(27,28);(2,7,11,9)(3,13)(8,22,23,20)(10,21,26,24)(12,27,25,29)(14,28)(15,18)(30,32);(1,3,4,14,6,13,17,28)(2,8,9,22,11,23,7,20)(5,18,16,32,19,15,31,30)(10,25,24,29,26,12,21,27);(10,26)(12,25)(21,24)(27,29)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231703, #72250212, C2×#512160 |
| maximal normal subgroup of | #72231703, #72250212, C2×#512160 |
| maximal quotient of | #72257229, #72257231, #72314009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419356);