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 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,12)(7,11,8,10)(14,18)(15,20)(16,19);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,18)(15,20)(16,19);(13,14,16,18,17,15,19,20);(2,4)(13,15,16,20,17,14,19,18);(5,8,9,7)(6,11,12,10)(14,15)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257098, #72257122, #72257144, … |
| maximal normal subgroup of | #72257098, #72257122, #72257144, … |
| maximal quotient of | #72264564, #72308690, #72319895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470822);