Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1505 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7580 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,6,13,22,29,8,14,23,11,7,10,4,20)(3,15,16,9,18,26,21,27,19,31,24,28,17,30,32,25);(1,3)(2,9)(4,21)(5,24)(6,18)(7,17)(8,19)(10,25)(11,30)(12,26)(13,27)(14,28)(15,20)(16,23)(22,32)(29,31);(2,10,14,13)(4,22)(5,23)(9,25,28,27)(11,20,12,29)(15,26,31,30)(16,24)(21,32)Polycyclic presentation:
2^9|a^2=e,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31568);