Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(7,11)(9,12)(13,14,16,18,17,20,15,19);(6,9,10,12);(2,4)(5,6)(7,12)(8,10)(9,11)(14,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,e^c=ei,f^c=fh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213145, #72213184, #72213342, … |
| maximal normal subgroup of | #72213145, #72213342, #72213348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23597);