Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | D4×#135 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,3,8,6)(10,15,16,23)(11,19)(12,13)(17,20,18,24)(21,22);(5,8)(6,7)(9,10)(11,20)(12,18)(13,17)(14,16)(15,21)(19,23)(22,24);(1,3)(2,4)(5,6)(7,8)(9,10)(11,20)(12,18)(13,17)(14,16)(15,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=g,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225271, #72227032, #72227132, … |
| maximal normal subgroup of | #72225271, #72227032, #72227132, … |
| maximal quotient of | #72225229, #72225231, #72225233 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39061);