Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,21,24,19)(15,20,23,28,16,25,26,27);(2,4)(14,19)(15,23)(16,26)(17,24)(21,22)(27,28);(6,9)(10,12)(13,15,17,23,18,16,24,26)(14,20,22,28,21,25,19,27);(13,16,17,26,18,15,24,23)(14,20,22,28,21,25,19,27);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,25)(27,28)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280255, #72284359, #72284557, … |
| maximal normal subgroup of | #72280255, #72284359, #72284557, … |
| maximal quotient of | #72311751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374684);