Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,15)(12,22,23,16)(13,18)(14,17)(19,31,32,26)(20,28)(21,27)(24,35,36,29)(25,30)(33,39,40,37)(34,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,14,21,25,34)(10,15,18,28,17,27,30,38)(12,19,24,33,23,32,36,40)(16,26,29,37,22,31,35,39);(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,22)(18,29)(20,33)(21,32)(25,36)(27,31)(28,37)(30,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^b=fh,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317370, #72317376, #72317394, … |
| maximal normal subgroup of | #72317370, #72317376, #72317394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13342);