Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152017 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12687 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1403 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 1403 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,5,10,6,11,21,28)(3,12,15,27,14,22,30,31)(4,8,18,25,17,9,20,26)(7,19,24,13,23,29,32,16);(1,3)(2,7)(4,13)(5,19)(6,14)(8,22)(9,12)(10,27)(11,23)(15,18)(16,17)(20,30)(21,29)(24,25)(26,32)(28,31);(1,4,6,17)(2,8,11,9)(3,13,14,16)(5,18,21,20)(7,22,23,12)(10,25,28,26)(15,29,30,19)(24,31,32,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,e^b=ei,f^b=fgi,g^b=gi,h^b=hi,d^c=di,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213638, #72213674, #72213697, … |
| maximal normal subgroup of | #72213638, #72213674, #72213697, … |
| maximal quotient of | C2×#152020, #72338268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59216);