Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152017 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12687 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1372 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 1372 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)(3,12,15,7)(4,8)(5,10,19,9)(6,11)(13,27,29,32)(14,25,18,21)(16,23)(17,22,30,24)(20,28,26,31);(1,3)(2,7)(4,13)(5,18)(6,14)(8,20)(9,12)(10,25)(11,21)(15,19)(16,28)(17,31)(22,27)(23,32)(24,26)(29,30);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,24)(10,22)(11,23)(12,26)(14,28)(15,29)(18,31)(19,30)(21,32)(25,27)Polycyclic presentation:
2^9|a^2=f,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 | #72213519, #72213618, #72213671, … |
| maximal normal subgroup of | #72213519, #72213618, #72213671, … |
| maximal quotient of | #72231264, #72240968, #72250644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59213);