Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #29634 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1079 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1079 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(22,25)(26,28);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,18,21,24)(19,25)(20,23,29,32)(22,27)(26,31)(28,30);(4,11)(6,12)(7,14)(9,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)Polycyclic presentation:
2^9|b^2=eh,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72240625, #72240629, #72240636, … |
| maximal normal subgroup of | #72240625, #72240629, #72240636, … |
| maximal quotient of | #72251448, #72251470, #72251472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159089);