Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8851 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,11,22,31,6,12,21,32,23,33,45,54)(3,13,15,34,17,35,40,55,16,36,39,56,41,57,59,64)(4,9,19,29,20,30,44,53,18,28,42,51,43,52,60,63)(8,24,26,38,27,47,50,37,25,46,48,58,49,61,62,14);(1,3,5,15,7,17,22,40,6,16,21,39,23,41,45,59)(2,8,10,26,11,27,31,50,12,25,32,48,33,49,54,62)(4,14,19,24,20,38,44,47,18,37,42,46,43,58,60,61)(9,13,29,34,30,35,53,55,28,36,51,56,52,57,63,64);(4,18)(14,37)(19,42)(20,43)(24,46)(38,58)(44,60)(47,61)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,e^b=egi,d^c=dg,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287454, #72343176, #72343182, … |
| maximal normal subgroup of | #72287454, #72343176, #72343182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43152);