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 | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,9,12,11)(13,14,16,20,17,18,23,25)(15,19,21,26,22,24,27,28);(1,2,4,3)(6,9)(7,12)(10,11);(1,3,4,2)(14,18)(19,24)(20,25)(26,28);(6,10)(9,11)(13,15,16,21,17,22,23,27)(14,19,20,26,18,24,25,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eh,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72318330, #72318410, C2×#184140, … |
| maximal normal subgroup of | #72318330, #72318410, C2×#184140, … |
| maximal quotient of | #72318636, C2×#184140, #72321323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91336);