Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10991 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 1622 conjugacy classes of subgroups — too many to draw. Only its 136 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:
(1,2,7,13,8,14,29,42)(3,15,19,43,20,33,28,53)(4,10,23,37,24,38,52,63)(5,12,25,41,6,11,27,39)(9,30,34,18,32,47,57,17)(16,35,44,59,45,58,54,61)(21,48,49,64,46,31,55,56)(22,36,50,60,26,40,51,62);(1,3,8,20)(2,9,14,32)(4,21,24,46)(5,18,6,17)(7,28,29,19)(10,35,38,58)(11,33,12,15)(13,34,42,57)(16,26,45,22)(23,49,52,55)(25,30,27,47)(31,40,48,36)(37,61,63,59)(39,53,41,43)(44,51,54,50)(56,60,64,62);(1,4)(2,10)(3,16)(5,22)(6,26)(7,23)(8,24)(9,31)(11,36)(12,40)(13,37)(14,38)(15,35)(17,21)(18,46)(19,44)(20,45)(25,50)(27,51)(28,54)(29,52)(30,49)(32,48)(33,58)(34,56)(39,60)(41,62)(42,63)(43,59)(47,55)(53,61)(57,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,f^c=fi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267974, #72267975, #72267981, … |
| maximal normal subgroup of | #72267974, #72267975, #72267981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54965);