Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106017 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1118 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)(11,19)(12,23,24,33)(13,17)(14,18)(15,21)(16,25,31,27)(20,36,38,40)(22,32)(26,30)(28,34)(29,39,35,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,18,19)(12,20)(13,26)(14,21)(16,29,31,35)(17,32,34,30)(22,28)(23,36,33,40)(24,38)(25,39)(27,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,33)(18,31)(19,35)(21,38)(22,39)(23,34)(25,28)(26,37)(30,36)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280963 |
| maximal normal subgroup of | #72280963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13213);