Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,14,17)(11,15,21,26)(12,16,23,28)(13,18,24,30)(19,31,33,25)(20,27,34,37)(22,29,35,38)(32,39,40,36);(3,4)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,32)(23,33)(24,34)(28,31)(29,36)(30,37)(35,40)(38,39);(1,3,5,4)(2,6,8,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,28)(18,29)(20,32)(21,33)(24,35)(26,31)(27,36)(30,38)(34,40)(37,39);(1,4,5,3)(2,7,8,6)(10,17)(15,26)(16,28)(18,30)(25,31)(27,37)(29,38)(36,39);(2,8)(6,7)(9,13,14,24)(10,18,17,30)(11,20,21,34)(12,22,23,35)(15,27,26,37)(16,29,28,38)(19,32,33,40)(25,36,31,39)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311122, #72311148, #72311177, … |
| maximal normal subgroup of | #72311122, #72311148, #72311177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394850);