Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,13,15,16)(17,18,19,25,21,26,27,22,20,23,28,24,29,31,32,30);(2,5)(6,7)(9,10,14,13,12,15,11,16)(18,22,23,30)(19,27)(21,29)(24,26,25,31)(28,32);(1,3,4,8)(2,6,5,7)(18,23)(22,30)(24,25)(26,31)Polycyclic presentation:
2^9|a^2=f,b^2=defg,c^2=ei,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343348, #72343363, #72343365, … |
| maximal normal subgroup of | #72343348, #72343363, #72343365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49056);