Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator | C22 | C22×C4×C8 |
| socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1542 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)(13,14,17,16,18,20,25,24)(15,19,23,21,22,26,28,27);(2,4)(5,6,7,9,8,10,11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(13,16,17,20,18,24,25,14)(15,21,23,26,22,27,28,19);(15,22)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^b=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72224770, #72224773, #72224774, … |
| maximal normal subgroup of | #72224770, #72224773, #72224774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89084);