Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14,19,22)(15,17,21,18)(16,24,27,23)(20,25,28,26);(1,2,3,4)(5,7,8,12,9,6,10,11)(13,15,17,22,19,21,18,14)(16,24,25,20,27,23,26,28);(13,16,18,26,19,27,17,25)(14,20,21,24,22,28,15,23);(6,7)(11,12)(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,24,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,d^b=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224967, #72225100, #72225109, … |
| maximal normal subgroup of | #72224967, #72225100, #72225109, … |
| maximal quotient of | #72224956, #72287494, C2×#204364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111560);