Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152322 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12852 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#535 | C22 |
| 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 | |
598 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 598 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,12,14,7)(4,16,18,24)(5,11,20,10)(6,9)(8,17,25,19)(13,27,29,26)(15,22)(21,28,32,30)(23,31);(1,3,6,14)(2,7,9,12)(4,13,17,28)(5,15,20,31)(8,21,24,26)(10,22,11,23)(16,32,25,27)(18,30,19,29);(1,4)(2,8,9,24)(3,13)(5,19)(6,17)(7,21,12,26)(10,25,11,16)(14,28)(15,29)(18,20)(22,27,23,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=d,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,f^b=fi,e^c=ei,f^c=fgi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213519, #72213622, #72213697, … |
| maximal normal subgroup of | #72213519, #72213622, #72213697, … |
| maximal quotient of | #72245418, #72250646, C2×#152323 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59519);