Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,20,27,37,16,21,13,29,12,18,28,35)(11,22,25,33,23,36,31,34,26,39,24,38,32,40,30,17);(9,11)(10,17,21,34)(12,23)(13,30)(14,31)(15,32)(16,26)(18,33,20,38)(19,36,29,40)(22,35,39,37)(24,28)(25,27);(2,5)(6,8)(9,12,16,15)(10,18,21,20)(11,23,26,32)(13,27,14,28)(17,33,34,38)(19,35,29,37)(22,36,39,40)(24,30,25,31);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72310113, #72310115, #72310118, … |
| maximal normal subgroup of | #72310113, #72310115, #72310118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402724);