Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403338 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14873 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 280 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 — 4 generators on 32 points:
(1,2)(3,13)(4,17)(5,12)(6,10)(7,11)(8,15)(9,19)(14,28)(16,27)(18,25)(20,26)(21,23)(22,29)(24,31)(30,32);(1,3)(2,8,11,23)(4,14)(5,15)(6,21)(7,16)(9,24,17,22)(10,13,12,27)(18,29)(19,30)(20,31)(25,32,26,28);(1,4)(2,9,11,17)(3,14)(5,18)(6,20)(7,19)(8,22,23,24)(10,25,12,26)(13,28,27,32)(15,29)(16,30)(21,31);(2,10)(5,6)(8,13)(9,25)(11,12)(15,21)(17,26)(18,20)(22,28)(23,27)(24,32)(29,31)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| Fitting subgroup of | #72233782 |
| maximal subgroup of | #72233772, #72233782, #72319564, … |
| maximal normal subgroup of | #72233772, #72233782, #72319564, … |
| maximal quotient of | #72319662, #72319738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310534);