Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #34609 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,32)(30,31);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(23,31)(26,32);(4,14)(7,12)(9,16)(10,13)(17,19)(18,23)(20,27)(21,30)(22,28)(24,32)(25,31)(26,29);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279475, #72279844, #72284395, … |
| maximal normal subgroup of | #72279475, #72279844, #72284395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142631);