Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#459 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,11,13,12,15,16,14)(17,18,19,23,21,24,20,22);(1,2,6,8,4,7,3,5)(10,13,15,14)(18,22)(19,20)(23,24);(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216350, #72216473, #72216495 |
| maximal normal subgroup of | #72216350, #72216473, #72216495 |
| maximal quotient of | #72216313, #72216471, #72225445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053);