Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94172 |
| 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:
(1,2,5,10,7,12,25,37)(3,13,15,41,17,33,23,53)(4,18,20,49,22,9,32,34)(6,11,26,38,27,39,24,36)(8,28,29,19,31,45,56,14)(16,42,47,64,48,60,46,63)(21,40,52,61,43,35,51,62)(30,57,58,50,59,54,55,44);(1,3,6,16,7,17,27,48)(2,8,11,30,12,31,39,59)(4,19,21,50,22,14,43,44)(5,23,24,47,25,15,26,46)(9,33,35,60,18,13,40,42)(10,29,36,55,37,56,38,58)(20,45,51,57,32,28,52,54)(34,41,61,63,49,53,62,64)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=efgi,f^a=fh,g^a=ghi,c^b=ci,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, 1371);