Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8282 |
| prime radical | C23 | #443 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,13,16,34,17,41,25,55)(4,9)(5,10)(6,12)(7,11)(8,15,28,22,29,44,38,46)(14,40,43,62,18,45,48,58)(19,31)(20,33)(21,32)(23,36)(24,35)(26,37)(27,42,56,53,30,47,57,49)(39,54)(50,60)(51,59)(52,61)(63,64);(1,3)(2,8,11,29)(4,18,20,14)(5,22)(6,25)(7,17)(9,30)(10,34,35,55)(12,38,37,28)(13,39,41,36)(15,23)(16,26)(19,49,50,53)(21,43,52,48)(24,46)(27,33)(31,58)(32,56)(40,63)(42,51,47,64)(44,54)(45,59)(57,61)(60,62);(1,4,5,19,6,21,23,51,7,20,24,50,26,52,54,64)(2,9,10,31,12,32,36,59,11,33,35,60,37,61,39,63)(3,14,22,53,25,48,15,42,17,18,46,49,16,43,44,47)(8,27,34,62,38,57,13,40,29,30,55,58,28,56,41,45)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,g^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,g^b=gi,h^b=hi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320434, #72320453, #72320456, … |
| maximal normal subgroup of | #72320434, #72320453, #72320456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36347);