Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,14,20)(8,16)(9,12)(10,18)(13,19,17,15);(2,4)(5,7)(6,11)(8,17)(9,14)(10,15)(12,20)(13,18)(16,19);(5,8,16,6,9,18,10,12)(7,13,15,11,14,17,19,20);(2,4)(5,7)(6,11)(8,13)(9,14)(10,19)(12,20)(15,16)(17,18)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219647, #72219653, #72219668, … |
| maximal normal subgroup of | #72219647, #72219653, #72219668, … |
| maximal quotient of | #72232722, #72233123, #72260439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281230);