Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144075 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1695 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10278 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,18,23,48)(4,24)(5,14)(6,13)(7,16)(8,15)(9,17)(10,20,38,56)(11,26)(12,32)(19,50,54,64)(21,49,55,37)(22,45,34,36)(25,58,52,39)(27,43)(28,47)(29,41)(30,40)(31,42)(33,46)(35,57,62,53)(44,60)(51,63,59,61);(1,3)(2,10)(4,25)(5,21)(6,20)(7,34)(8,22)(9,23)(11,39)(12,45)(13,36)(14,48)(15,49)(16,37)(17,38)(18,46)(19,27)(24,50)(26,53)(28,51)(29,59)(30,52)(31,54)(32,56)(33,55)(35,41)(40,64)(42,63)(43,61)(44,58)(47,62)(57,60);(1,4)(2,11)(3,19,23,54)(5,31)(6,26)(7,29)(8,28)(9,30)(10,35,38,62)(12,41)(13,47)(14,24)(15,43)(16,42)(17,44)(18,39,48,58)(20,52,56,25)(21,51,55,59)(22,57,34,53)(27,33)(32,60)(36,63,45,61)(37,64,49,50)(40,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51271);