Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| 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 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(1,2,4,7,5,8,3,6)(10,14,17,16)(15,22,18,23);(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72342986, #72342988, #72343011, … |
| maximal normal subgroup of | #72342986, #72342988, #72343011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13324);