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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,24)(18,21)(19,22)(20,26)(23,28)(25,27);(1,2)(3,4)(6,9)(10,12)(13,15,19,21)(14,17,22,18)(16,25,28,20)(23,26,24,27);(5,7,8,11)(6,10,9,12)(13,16,18,27,19,28,17,26)(14,20,15,23,22,25,21,24);(2,4)(13,16,17,26,19,28,18,27)(14,20,21,24,22,25,15,23);(6,9)(10,12)(13,17,19,18)(14,15,22,21)(16,26,28,27)(20,23,25,24)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=g,e^2=i,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468336);