Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1667 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8644 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,7,14,9,16,8,15)(3,17,21,44,23,45,22,33)(4,24,26,54,27,55,31,42)(5,13,6,12,29,40,28,41)(10,20,34,51,36,50,35,19)(11,30,37,56,39,57,38,25)(18,46,48,59,49,58,53,32)(43,52,61,63,62,64,60,47);(1,3,6,20,7,21,29,51,9,23,28,50,8,22,5,19)(2,10,13,33,15,35,41,45,16,36,40,44,14,34,12,17)(4,18,30,52,26,48,56,63,27,49,57,64,31,53,25,47)(11,32,42,60,38,58,55,62,39,59,54,61,37,46,24,43);(1,4)(2,11)(3,18)(5,25)(6,30)(7,26)(8,31)(9,27)(10,32)(12,24)(13,42)(14,37)(15,38)(16,39)(17,43)(19,47)(20,52)(21,48)(22,53)(23,49)(28,57)(29,56)(33,60)(34,46)(35,58)(36,59)(40,54)(41,55)(44,61)(45,62)(50,64)(51,63)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=eh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225389, #72225399, #72225401, … |
| maximal normal subgroup of | #72225389, #72225399, #72225401, … |
| maximal quotient of | #72225459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42167);