Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4290 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)(11,15)(12,22,24,16)(13,17)(14,18)(19,31,33,26)(20,27)(21,28)(23,35,36,29)(25,30)(32,39,40,37)(34,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,31)(23,32)(24,33)(25,34)(29,37)(30,38)(35,39)(36,40);(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,22)(20,32)(21,33)(25,36)(27,37)(28,31)(30,35)(34,40)(38,39);(3,8)(5,7)(9,13,14,25)(10,17,18,30)(11,20,21,34)(12,23,24,36)(15,27,28,38)(16,29,22,35)(19,32,33,40)(26,37,31,39);(2,6)(5,7)(11,21)(15,28)(19,33)(20,34)(26,31)(27,38)(32,40)(37,39)Polycyclic presentation:
2^9|a^2=h,d^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303162, #72303173, #72303238, … |
| maximal normal subgroup of | #72303162, #72303173, #72303238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293172);