Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,16)(11,14,12,15)(17,18,21,19,20,22,24,23);(2,4)(10,14)(11,12)(15,16)(17,18,21,19,20,22,24,23);(5,7,8,6)(10,15)(11,12)(14,16)(17,19,21,22,20,23,24,18)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh,e^b=eh,d^c=dh,e^c=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72321438, #72321451, #72321474, … |
| maximal normal subgroup of | #72321438, #72321451, #72321474, … |
| maximal quotient of | #72322431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16757);