Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23898 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,13)(4,19)(5,23,14,10)(6,25)(7,12)(8,16,31,40)(9,21)(11,28)(15,42,45,63)(17,39)(18,34)(20,52,41,62)(22,36)(24,35,27,37)(26,33)(29,57)(30,47,56,49)(32,53,55,54)(38,44,58,48)(43,51)(46,60)(50,61)(59,64);(1,3,14,40)(2,8,23,13)(4,15,41,43)(5,16,7,18)(6,17,24,47)(9,29,55,38)(10,34,12,31)(11,30,35,61)(19,51,62,63)(20,46,22,45)(21,44,53,57)(25,56,37,39)(26,48,54,64)(27,49,28,50)(32,58,33,59)(36,60,52,42);(1,4)(2,9)(3,15)(5,20)(6,26)(7,22)(8,29)(10,32)(11,36)(12,33)(13,38)(14,41)(16,46)(17,48)(18,45)(19,25)(21,28)(23,55)(24,54)(27,53)(30,60)(31,59)(34,58)(35,52)(37,62)(39,63)(40,43)(42,61)(44,50)(47,64)(49,57)(51,56)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,e^d=ei,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46941);