Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9467 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(10,15)(12,13)(16,23)(17,18)(19,21)(22,24);(2,6,8,7)(9,10,13,18,14,17,12,15)(11,16,19,23,20,24,21,22);(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=ei,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72280491, #72280517, #72280607, … |
| maximal normal subgroup of | #72280491, #72280517, #72280607, … |
| maximal quotient of | #72287450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15412);