Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | #97 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,13,7,18,20)(4,8,15,21,16,22,28,31)(12,24,27,32,26,19,29,30)(33,34)(35,45,48,56)(36,42)(37,43)(38,41)(39,44)(40,47,53,46)(49,59,62,64)(50,57)(51,55)(52,58)(54,61,63,60);(1,3)(2,7,10,11)(4,12)(5,14)(6,13)(8,19,22,24)(9,20,23,25)(15,27)(16,26)(17,18)(21,30,31,32)(28,29)(33,35,37,49,39,47,52,61,38,48,51,62,36,46,50,60)(34,40,43,54,44,45,58,59,41,53,55,63,42,56,57,64);(1,4,5,15,6,16,17,28)(2,8,9,21,10,22,23,31)(3,12,14,27,13,26,18,29)(7,19,20,30,11,24,25,32)(34,41)(35,46,48,47)(37,51)(40,45,53,56)(42,44)(49,61,62,60)(50,52)(54,64,63,59)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35637);