Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10,14,18,13,17,15,19)(11,16,21,24,12,22,20,23);(5,6)(7,8)(9,11,15,21,13,12,14,20)(10,16,19,24,17,22,18,23);(1,2,3,4)(6,8)(9,12,13,11)(10,16,17,22)(14,20,15,21)(18,24,19,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238743, #72238745, #72238750, … |
| maximal normal subgroup of | #72238743, #72238745, #72238750, … |
| maximal quotient of | #72239044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31154);