Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152069 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12694 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 710 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)(3,12,15,21)(4,8)(5,10,19,9)(6,11)(7,14,22,18)(13,27,29,20)(16,24)(17,23,30,25)(26,31,32,28);(1,3,6,14)(2,7)(4,13,16,28)(5,18,19,15)(8,20)(9,12)(10,22)(11,21)(17,31,30,29)(23,27)(24,32)(25,26);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,25)(10,23)(11,24)(12,26)(14,28)(15,29)(18,31)(19,30)(21,32)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=hi,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=fgh,g^b=gi,d^c=di,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213428, #72216577, #72216589, … |
| maximal normal subgroup of | #72213428, #72216577, #72216589, … |
| maximal quotient of | C2×#152069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59265);