Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #466 | C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(3,9)(4,11)(7,14)(8,15)(10,13)(12,16)(17,18,20,25,22,26,33,40,21,24,32,37,34,38,45,47)(19,27,29,39,30,41,43,48,28,23,31,35,42,36,44,46);(1,2)(3,10)(4,13)(5,7)(6,9)(8,11)(12,15)(14,16)(17,19)(18,23)(20,31)(21,28)(22,30)(24,27)(25,39)(26,36)(29,32)(33,44)(34,42)(35,37)(38,41)(40,48)(43,45)(46,47);(2,7)(6,14)(8,13)(10,15)(18,24)(19,28)(25,37)(26,38)(29,31)(30,42)(40,47)(43,44)Polycyclic presentation:
2^9|a^2=e,d^2=fhi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=chi,d^b=dfi,e^b=ei,f^b=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29703);