Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,18,23,19,22,26,21,25)(20,24,28,27,29,31,30,32);(6,8)(10,13)(14,15)(17,19,23,26,22,25,21,18)(20,27,28,31,29,32,30,24);(2,4)(5,6,7,8)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,32)(26,31);(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,25,26,19)(20,28,29,30)(24,27,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264168, #72264176, #72264277, … |
| maximal normal subgroup of | #72264168, #72264176, #72264277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275447);