Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(1,2)(3,7)(4,6)(5,8)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(3,8)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ei,f^b=fhi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214049, #72214640, #72214761, … |
| maximal normal subgroup of | #72214049, #72214640, #72214761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14329);