Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(18,23)(20,21)(24,31)(25,26)(27,32)(29,30);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,24)(20,30)(21,29)(22,28)(23,31)(25,27)(26,32);(19,28)(24,32)(27,31)(29,30)Polycyclic presentation:
2^9|b^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72310664, #72310683, #72310686, … |
| maximal normal subgroup of | #72310664, #72310683, #72310686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142259);