Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#95 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 862 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,5,4,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(10,13)(11,16)(14,15)(17,19,21,27)(18,22,23,30)(20,28,29,26)(24,32,31,25);(10,14)(13,15)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=e,b^2=gi,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fi,d^b=dg,e^b=efg,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72293071, #72293138, #72293146, … |
| maximal normal subgroup of | #72293071, #72293138, #72293146, … |
| maximal quotient of | #72293306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31746);