Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #392 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,21)(16,24)(17,20)(18,22)(19,25)(23,27)(26,28);(5,6,7,10,9,12,8,11)(13,15,16,22,17,23,25,28)(14,18,19,21,20,26,24,27);(6,10)(7,8)(11,12)(18,26)(21,27)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=ehi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72211654, C2×#118020, #72237225, … |
| maximal normal subgroup of | #72211654, C2×#118020, #72237225, … |
| maximal quotient of | #72211626, C2×#118020, #72216508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25216);