Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130786 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61091 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
791 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 791 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,7,14,9,15,30,51,8,16,35,42,6,13,25,49)(3,17,21,53,23,45,58,37,22,54,60,63,20,52,34,61)(4,24,27,46,26,44,5,12,28,47,32,48,33,11,31,43)(10,36,39,59,40,18,50,56,41,55,62,19,38,57,64,29);(1,3,8,22)(2,10,16,41)(4,18,28,57)(5,29,31,56)(6,23,9,20)(7,34,35,58)(11,37,44,61)(12,45,43,52)(13,40,15,38)(14,50,42,64)(17,46,54,48)(19,27,59,32)(21,30,60,25)(24,53,47,63)(26,36,33,55)(39,49,62,51);(1,4)(2,11)(3,18)(5,30)(6,33)(7,27)(8,28)(9,26)(10,37)(12,14)(13,47)(15,24)(16,44)(17,39)(19,58)(20,36)(21,56)(22,57)(23,55)(25,31)(29,60)(32,35)(34,59)(38,53)(40,63)(41,61)(42,43)(45,50)(46,49)(48,51)(52,64)(54,62)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307218, #72307293, #72307547, … |
| maximal normal subgroup of | #72307218, #72307293, #72307547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37984);