Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14,16,20)(15,18,24,26)(17,25,23,21)(19,28,27,22);(1,3,4,2)(6,10)(7,8)(11,12)(14,18)(15,22)(17,23)(19,21)(20,26)(24,28)(25,27);(6,11)(7,8)(10,12)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225208, D8×#367, #72226892 |
| maximal normal subgroup of | #72225208, D8×#367, #72226892 |
| maximal quotient of | #72225193, #72225602, #72225603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43527);