Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| 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 | C2×#135 | C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
272 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,18)(19,28,31,45)(20,24)(21,35)(22,27)(23,30,39,47)(25,33)(26,34)(29,44,32,41)(36,43,48,38)(37,40)(42,46);(17,19)(18,23)(20,29)(21,36)(22,31)(24,38)(25,41)(26,43)(27,39)(28,40)(30,46)(32,34)(33,47)(35,45)(37,48)(42,44);(2,6)(7,12)(8,13)(14,16)(17,20,22,34)(18,24,27,26)(19,29,31,32)(21,33,37,46)(23,38,39,43)(25,40,42,35)(28,44,45,41)(30,36,47,48)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320334, #72320381, #72320404, … |
| maximal normal subgroup of | #72320334, #72320381, #72320404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29672);