Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #95118 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(5,7)(6,9,10,11)(8,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(2,4)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257488, #72257491, #72257567, … |
| maximal normal subgroup of | #72257488, #72257491, #72257567, … |
| maximal quotient of | #72258986, #72264364, #72266885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12901);