Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9469 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,6,12)(3,14,18,43)(4,20,15,45)(5,11)(7,28,21,13)(8,29,32,16)(9,25)(10,36,38,23)(17,42)(19,51,46,44)(22,53)(24,40)(26,52)(27,35)(30,49)(31,47)(33,48,41,58)(34,55)(37,57)(39,54)(50,61)(56,62)(59,63)(60,64);(1,3,15,29,21,46,23,48,6,18,4,16,7,19,36,58)(2,8,10,14,28,41,45,51,12,32,38,43,13,33,20,44)(5,24,26,47,54,50,55,64,25,17,22,49,27,56,57,63)(9,30,37,40,39,60,53,62,11,31,34,42,35,59,52,61);(2,9)(5,25)(8,30)(10,37)(11,12)(13,35)(14,40)(17,24)(20,52)(22,26)(27,54)(28,39)(31,32)(33,59)(34,38)(41,60)(42,43)(44,61)(45,53)(47,49)(50,56)(51,62)(55,57)(63,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72321074, #72342989, #72343007 |
| maximal normal subgroup of | #72321074, #72342989, #72343007 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46775);