Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139736 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #3487 | C22 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 662 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)(6,11,28,39)(7,12)(8,16,31,42)(9,21,33,26)(15,44)(17,41,25,56)(18,34)(20,53,43,52)(22,37)(24,38)(27,36)(29,57)(30,50)(32,55)(35,54)(40,46,58,49)(45,60,63,51)(47,62)(48,61)(59,64);(1,3,14,42)(2,8,23,13)(4,15,43,63)(5,16,7,18)(6,25,27,48)(9,29,32,40)(10,34,12,31)(11,30,38,41)(17,24,50,28)(19,51,53,44)(20,45,22,47)(21,49,55,57)(26,46,54,64)(33,59,35,58)(36,56,39,61)(37,60,52,62);(1,4,7,22)(2,9,12,33)(3,15,18,47)(5,20,14,43)(6,26,28,21)(8,29,31,59)(10,35,23,32)(11,37,39,19)(13,40,34,58)(16,45,42,63)(17,49,25,46)(24,55,27,54)(30,60,61,51)(36,53,38,52)(41,62,56,44)(48,64,50,57)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72303086, #72303148, #72303150, … |
| maximal normal subgroup of | #72303086, #72303148, #72303150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46933);