Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32926 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,11)(4,15)(5,8)(6,10)(7,12)(9,17)(13,23)(14,24)(16,28)(18,22)(19,25)(20,26)(21,29)(27,31)(30,32);(1,3,4,12,6,14,17,26)(2,7,9,11,10,20,15,24)(5,13,16,25,18,27,29,32)(8,19,21,23,22,30,28,31);(2,8)(7,19)(9,21)(10,22)(11,23)(15,28)(20,30)(24,31);(5,18)(13,27)(16,29)(25,32)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,g^c=ghi,f^d=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231612, #72231703, #72231708, … |
| maximal normal subgroup of | #72231612, #72231703, #72231708, … |
| maximal quotient of | #72257211, C2×#511930, #72312861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419126);