Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,14)(10,13)(15,27)(16,23)(17,22)(18,25)(19,24)(20,26)(21,29)(28,35)(30,33)(31,34)(32,36);(1,3,4,2)(5,7)(6,11)(8,15)(9,16)(10,17)(12,21)(13,22)(14,23)(18,28)(19,29)(20,30)(24,27)(25,32)(26,33)(31,36)(34,35);(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,24)(14,25)(16,28)(17,29)(20,31)(22,27)(23,32)(26,34)(30,36)(33,35);(6,13)(11,22)(12,24)(14,26)(21,27)(23,33)(25,34)(32,35);(5,9,10,20)(6,14,13,26)(7,16,17,30)(8,18,19,31)(11,23,22,33)(12,25,24,34)(15,28,29,36)(21,32,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,g^b=gi,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72280674, #72280714, #72280720, … |
| maximal normal subgroup of | #72280674, #72280714, #72280720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278532);