Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(2,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(10,13)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(6,8)(9,11,12,15)(10,14,13,16)(19,28)(23,32)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=ef,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264202, #72264220, #72264309, … |
| maximal normal subgroup of | #72264202, #72264220, #72264309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311479);