Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535641 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10,13,11)(12,14,16,15)(18,23)(19,24)(20,21);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,18,21,19,22,24,20,23);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,19,20,18,22,23,21,24);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(18,24)(19,23);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72311908, #72312127 |
| maximal normal subgroup of | #72311908, #72312127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465015);