Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139572 |
| 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 | #525 | 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 | |
309 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 309 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:
(1,2,6,11)(3,13,15,42)(4,19,14,44)(5,23)(7,28,20,12)(8,29,32,16)(9,27)(10,36,38,22)(17,39)(18,50,45,43)(21,53)(24,41)(25,51)(26,35)(30,46)(31,48)(33,47,40,58)(34,54)(37,57)(49,62)(52,55)(56,61)(59,64)(60,63);(1,3,14,29,20,45,22,47,6,15,4,16,7,18,36,58)(2,8,10,13,28,40,44,50,11,32,38,42,12,33,19,43)(5,24,25,46,52,49,54,64,27,17,21,48,26,56,57,63)(9,31,37,41,55,59,53,61,23,30,34,39,35,60,51,62);(2,9)(3,15)(8,30)(10,37)(11,23)(12,35)(13,39)(16,29)(17,24)(18,45)(19,51)(28,55)(31,32)(33,59)(34,38)(40,60)(41,42)(43,61)(44,53)(46,48)(47,58)(49,56)(50,62)(63,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72342959, #72343002, #72343007 |
| maximal normal subgroup of | #72342959, #72343002, #72343007 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46774);