Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622976 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1706 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)(3,6)(4,5)(7,8)(10,16)(11,21)(12,13)(14,25)(18,29)(19,30)(20,34)(22,35)(23,24)(26,27)(28,32)(33,39)(36,37)(38,40);(2,5)(6,8)(9,10,13,19,14,20,26,29,15,18,27,34,25,30,12,16)(11,17,21,32,24,33,36,38,22,31,35,40,37,39,23,28);(1,3,4,7)(2,6,5,8)(9,11,12,23,25,37,27,35,15,22,26,36,14,24,13,21)(10,17,16,28,30,39,34,40,18,31,29,38,20,33,19,32);(11,22)(17,31)(21,35)(23,36)(24,37)(28,38)(32,40)(33,39);(10,18)(16,29)(17,31)(19,34)(20,30)(28,38)(32,40)(33,39)Polycyclic presentation:
2^9|b^2=g,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264975, #72264984, #72265058, … |
| maximal normal subgroup of | #72264975, #72264984, #72265058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530370);