Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,21,18,17,16,24)(15,20,22,28,23,26,25,27);(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,23)(19,22)(21,28)(24,27);(1,2,3,4)(13,16,18,19)(14,21,17,24)(15,22,23,25)(20,27,26,28);(13,17,19,24,18,14,16,21)(15,20,22,28,23,26,25,27);(5,7,8,11)(6,10,9,12)(14,17)(20,26)(21,24)(27,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=di,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72257335, #72280333, #72282160, … |
| maximal normal subgroup of | #72257335, #72280333, #72282160, … |
| maximal quotient of | #72257840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281106);