Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,20,29,18)(11,16,23,33,24,34,39,32)(12,25,13,17,26,35,28,36)(21,37,27,30,38,40,22,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,27)(14,23)(15,24)(17,30)(18,32)(19,33)(20,34)(22,28)(25,37)(26,38)(29,39)(31,36)(35,40);(9,12,14,13,15,26,29,28)(10,17,18,25,20,36,19,35)(11,21,23,27,24,38,39,22)(16,30,32,37,34,31,33,40);(3,8)(5,7);(2,6)(5,7)(9,13,14,26,15,28,29,12)(10,17,19,35,20,36,18,25)(11,22,23,21,24,27,39,38)(16,31,33,37,34,30,32,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72264915, #72264923, #72265051, … |
| maximal normal subgroup of | #72264915, #72264923, #72265051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310587);