Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3487 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,16,11,15,9,14)(7,12,20,28,19,27,18,26)(8,21,22,30,23,13,24,29)(17,31,32,36,33,25,34,35);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,20)(11,19)(13,25)(14,26)(15,27)(16,28)(21,31)(22,32)(23,33)(24,34)(29,35)(30,36);(5,8,11,23)(6,13,15,21)(7,17,19,33)(9,22,10,24)(12,25,27,31)(14,29,16,30)(18,32,20,34)(26,35,28,36);(2,4)(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,27,28)(13,29,21,30)(17,32,33,34)(25,35,31,36);(7,19)(12,27)(17,33)(18,20)(25,31)(26,28)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311324, #72341700, #72341707 |
| maximal normal subgroup of | #72311324, #72341700, #72341707 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286224);