Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,4,9)(3,13,17,30,16,28,14,29)(5,19,18,27,20,10,21,25)(8,22,23,15,24,31,26,32);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,23)(10,13)(11,26)(12,24)(18,31)(19,28)(20,32)(21,22)(25,29)(27,30);(10,19)(13,28)(25,27)(29,30);(1,4,7,6)(2,9,12,11)(3,14,16,17)(5,18,20,21)(8,23,24,26)(10,25,19,27)(13,29,28,30)(15,31,32,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233438, #72234211, #72234212, … |
| maximal normal subgroup of | #72233438, #72234211, #72234212, … |
| maximal quotient of | #72238629, #72251987, #72260446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400765);