Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94168 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7213 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #2310 | C22 |
| prime radical | C24 | #95 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G | |
| LECS | G ▷ #2310 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 64 points:
(1,2,5,10,7,12,24,36)(3,13,15,38,17,40,22,53)(4,18,19,48,21,9,32,33)(6,25,26,35,27,11,23,37)(8,28,29,42,31,43,57,14)(16,45,46,61,47,39,44,62)(20,34,51,60,52,49,50,59)(30,55,58,41,54,63,56,64);(1,3,6,16,7,17,27,47)(2,8,11,30,12,31,25,54)(4,14,20,41,21,42,52,64)(5,22,23,46,24,15,26,44)(9,13,34,39,18,40,49,45)(10,29,35,56,36,57,37,58)(19,28,50,55,32,43,51,63)(33,53,59,62,48,38,60,61)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=i,e^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=ghi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1367);