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 | #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 | |
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:
(1,2,6,12)(3,14,16,42)(4,20,15,44)(5,11)(7,28,21,13)(8,29,31,17)(9,25)(10,36,38,23)(18,40)(19,50,45,43)(22,52)(24,55)(26,51)(27,35)(30,46)(32,47,41,58)(33,48)(34,54)(37,57)(39,53)(49,64)(56,61)(59,63)(60,62);(1,3,15,29,21,45,23,47,6,16,4,17,7,19,36,58)(2,8,10,14,28,41,44,50,12,31,38,42,13,32,20,43)(5,24,26,46,53,49,54,63,25,18,22,48,27,56,57,62)(9,33,37,55,39,59,52,61,11,30,34,40,35,60,51,64);(2,9)(3,16)(5,25)(8,30)(10,37)(11,12)(13,35)(14,40)(17,29)(19,45)(20,51)(22,26)(27,53)(28,39)(31,33)(32,59)(34,38)(41,60)(42,55)(43,61)(44,52)(47,58)(50,64)(54,57)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,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72321074, #72342991, #72343002 |
| maximal normal subgroup of | #72321074, #72342991, #72343002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46776);