Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12695 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #60704 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3455 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11)(3,13,16,30)(4,8,18,22)(6,10,19,27)(7,12,21,29)(9,23,24,17)(14,26,20,32)(15,25,31,28);(2,8)(4,6)(5,19)(9,13)(10,12)(15,17)(16,23)(18,21)(20,31)(22,27)(25,26)(30,32);(1,3)(2,9,12,25)(4,15)(5,20)(6,17)(7,14)(8,13,10,26)(11,28,29,24)(16,21)(18,23)(19,31)(22,32,27,30)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,d^c=di,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72292696, C2×#152080 |
| maximal normal subgroup of | #72292696, C2×#152080 |
| maximal quotient of | C2×#152080, #72293065, #72293072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59276);