Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #466 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8596 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,14,8,15,5,30,9,16,18,12,6,13,31,42)(3,17,22,35,23,46,20,10,24,36,47,37,21,45,53,38)(4,25,28,57,27,41,58,43,29,40,33,59,32,11,26,39)(19,48,51,61,50,62,64,60,52,63,55,34,54,44,49,56);(1,3,18,47,8,23,31,53,9,24,7,22,6,21,5,20)(2,10,30,45,13,35,14,36,16,38,42,46,15,37,12,17)(4,19,28,51,27,50,58,64,29,52,33,55,32,54,26,49)(11,34,59,63,40,60,43,62,41,61,57,48,25,56,39,44);(1,4)(2,11)(3,19)(5,26)(6,32)(7,33)(8,27)(9,29)(10,34)(12,39)(13,40)(14,43)(15,25)(16,41)(17,44)(18,28)(20,49)(21,54)(22,55)(23,50)(24,52)(30,59)(31,58)(35,60)(36,62)(37,56)(38,61)(42,57)(45,63)(46,48)(47,51)(53,64)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320522, #72320529, #72320534, … |
| maximal normal subgroup of | #72320522, #72320529, #72320534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41879);