Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,23)(18,21)(19,22)(20,27)(24,25)(26,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,22,19,21,17,14)(16,24,27,20,25,23,26,28);(7,12)(9,10)(13,16,18,27,19,25,17,26)(14,20,15,23,22,28,21,24);(5,7)(6,10)(8,12)(9,11)(16,25)(20,28)(23,24)(26,27);(7,12)(9,10)(13,17,19,18)(14,15,22,21)(16,26,25,27)(20,23,28,24)Polycyclic presentation:
2^9|b^2=fh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257359, #72257612, #72257637, … |
| maximal normal subgroup of | #72257359, #72257612, #72257637, … |
| maximal quotient of | #72264595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514380);