Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3459 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,23,31,14,15,24,26,25,27,36,37)(11,19,20,32,21,33,34,40,22,16,28,29,35,30,38,39);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,15)(16,19)(17,26)(18,27)(29,32)(30,33)(31,37)(39,40);(2,6)(5,7);(9,11,12,20,13,21,23,34,14,22,24,28,25,35,36,38)(10,16,17,29,18,30,31,39,15,19,26,32,27,33,37,40)Polycyclic presentation:
2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72322106 |
| maximal normal subgroup of | #72322106 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529963);