Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,12)(9,11)(14,19)(15,17)(16,20);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,14,17,19,18,16,15,20);(5,8,9,10)(6,7,11,12)(13,15,18,17)(14,19,16,20);(2,4)(6,11)(7,12)(13,15,18,17)(14,20,16,19);(13,16,17,20,18,14,15,19)Polycyclic presentation:
2^9|b^2=fgi,c^2=g,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257098, #72257108, #72257158, … |
| maximal normal subgroup of | #72257098, #72257108, #72257158, … |
| maximal quotient of | #72308715, #72320263 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449137);