Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×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 | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,10,11)(7,9,12,8)(13,14,18,21,17,20,26,28)(15,22,24,25,23,19,27,16);(2,4)(5,7,9,11,10,12,8,6)(13,15)(14,19)(16,21)(17,23)(18,24)(20,22)(25,28)(26,27);(5,8,10,9)(6,7,11,12)(13,16,18,22,17,25,26,19)(14,15,21,24,20,23,28,27);(5,8,10,9)(6,7,11,12)(14,20)(15,23)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,d^b=dhi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286991 |
| maximal normal subgroup of | #72286991 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89735);