Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134702 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1809 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8600 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
254 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,7,14,6,15,32,41,9,16,31,12,8,13,5,30)(3,17,21,35,22,46,19,36,23,44,33,42,20,54,53,10)(4,24,27,57,28,40,59,38,29,43,58,56,26,11,25,39)(18,47,50,34,49,45,48,61,52,63,60,37,51,62,64,55);(1,3,5,19,8,20,31,21,9,23,32,53,6,22,7,33)(2,10,12,17,15,35,30,46,16,36,14,44,13,42,41,54)(4,18,25,48,26,51,58,50,29,52,59,64,28,49,27,60)(11,37,38,62,24,55,56,47,40,34,39,45,43,61,57,63);(1,4)(2,11)(3,18)(5,25)(6,28)(7,27)(8,26)(9,29)(10,34)(12,38)(13,43)(14,39)(15,24)(16,40)(17,45)(19,48)(20,51)(21,50)(22,49)(23,52)(30,56)(31,58)(32,59)(33,60)(35,61)(36,37)(41,57)(42,55)(44,62)(46,63)(47,54)(53,64)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cgh,e^b=ei,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217272, #72217274, #72224960, … |
| maximal normal subgroup of | #72217272, #72217274, #72224960, … |
| maximal quotient of | C2×#134702 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41898);