Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,6)(4,5)(7,8)(9,10,11,14,12,13,15,16)(17,18,21,22,23,26,20,25)(19,27,28,32,30,29,31,24);(2,5)(6,8)(17,19,22,29,21,31,26,32,23,30,25,27,20,28,18,24);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,18,20,25,23,26,21,22)(19,24,28,27,30,32,31,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,f^a=fi,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72343314, #72343320, #72343334, … |
| maximal normal subgroup of | #72343314, #72343320, #72343334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35407);