Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138574 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9215 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,7,13,30,11,29,12,5,15,16,41,8,14,6,31)(3,9,20,35,21,45,19,43,22,37,32,59,18,36,47,34)(4,23,26,38,24,42,48,40,28,44,58,10,27,39,25,46)(17,49,51,33,54,57,50,61,53,63,55,62,52,64,56,60);(1,3,16,47,30,18,6,32,5,22,7,19,8,21,29,20)(2,9,12,35,14,45,13,43,15,37,31,59,11,36,41,34)(4,17,48,51,27,54,26,50,28,53,25,55,24,52,58,56)(10,33,42,49,46,60,44,64,38,62,39,63,40,61,23,57);(1,4)(2,10)(3,17)(5,28)(6,26)(7,25)(8,24)(9,33)(11,40)(12,42)(13,44)(14,46)(15,38)(16,48)(18,54)(19,55)(20,56)(21,52)(22,53)(23,41)(27,30)(29,58)(31,39)(32,50)(34,57)(35,49)(36,61)(37,62)(43,64)(45,60)(47,51)(59,63)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=ghi,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45771);