Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,17,21,18,16,20,25)(15,19,24,27,23,22,26,28);(2,4)(5,6,8,12,9,11,7,10)(13,15)(14,19)(16,22)(17,24)(18,23)(20,26)(21,27)(25,28);(6,11)(10,12)(13,16,17,25,18,14,20,21)(15,22,24,28,23,19,26,27);(15,23)(19,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225125, #72285370, #72286025, … |
| maximal normal subgroup of | #72225125, #72285370, #72286025, … |
| maximal quotient of | #72225106, #72285724, #72285943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89691);