Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | |
2178 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)(14,17)(15,19)(18,20);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,14,15,17,16,18,19,20);(5,8,9,10)(6,7,11,12);(2,4)(6,11)(7,12)(14,18)(17,20);(13,14,19,20,16,18,15,17)Polycyclic presentation:
2^9|b^2=fg,c^2=g,e^2=fi,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^b=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257369, #72257580, #72263851, … |
| maximal normal subgroup of | #72257369, #72257580, #72263851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449191);