Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238834 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
962 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 962 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(17,19)(18,23)(20,28)(21,27)(22,29)(24,30)(25,31)(26,32);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(3,11)(7,10)(12,16)(14,15)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,30,32,31)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gi,d^c=dhi,f^c=fhi,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72305428, #72305433, #72305436, … |
| maximal normal subgroup of | #72305428, #72305433, #72305436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146034);