Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139572 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9468 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 489 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:
(1,2,6,12,7,14,22,13)(3,15,20,43,17,45,48,44)(4,21,24,10,16,39,47,40)(5,11,28,41,26,9,29,37)(8,30,32,46,34,58,33,18)(19,42,51,63,25,57,52,64)(23,54,55,38,27,53,56,36)(31,60,61,50,35,59,62,49);(1,3,16,46,22,48,24,30,7,17,4,18,6,20,47,58)(2,8,10,15,12,32,39,43,14,34,40,45,13,33,21,44)(5,25,27,49,28,52,56,60,26,19,23,50,29,51,55,59)(9,35,38,57,41,61,54,63,11,31,36,42,37,62,53,64);(2,9)(3,17)(5,26)(8,31)(10,38)(11,14)(12,41)(13,37)(15,42)(18,46)(20,48)(21,53)(23,27)(28,29)(30,58)(32,62)(33,61)(34,35)(36,40)(39,54)(43,64)(44,63)(45,57)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^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 | #72321514, #72342991, #72343002 |
| maximal normal subgroup of | #72321514, #72342991, #72343002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46772);