Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,20)(14,18)(15,19)(17,26)(21,36)(22,30)(23,32)(25,35)(27,31)(28,34)(29,38)(33,39)(37,40);(2,5)(6,8)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,33)(18,30)(19,32)(20,31)(24,36)(25,37)(26,39)(28,38)(29,34)(35,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,26)(15,25)(16,29)(18,35)(19,34)(20,24)(22,39)(23,37)(27,38)(30,36)(31,40)(32,33);(10,18,19,20)(12,25)(16,30,32,31)(17,24,34,35)(21,37)(26,28)(29,40,33,36)(38,39)Polycyclic presentation:
2^9|c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279937 |
| maximal normal subgroup of | #72279937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311416);