Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342866 |
| 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×#3533 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3326 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)(12,16,23,29)(13,17)(14,18)(19,25,32,36)(20,26)(21,27)(22,28,34,38)(24,30)(31,35,39,40)(33,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(9,12)(10,16,18,29)(11,19)(13,22)(14,23)(15,25,27,36)(17,28,30,38)(20,31)(21,32)(24,34)(26,35,37,40)(33,39);(3,8)(5,7)(9,13,14,24)(10,17,18,30)(11,20,21,33)(12,22,23,34)(15,26,27,37)(16,28,29,38)(19,31,32,39)(25,35,36,40);(2,6)(5,7)(11,21)(15,27)(19,32)(20,33)(25,36)(26,37)(31,39)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303107, #72303121, #72303183, … |
| maximal normal subgroup of | #72303107, #72303121, #72303183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300724);