Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14)(15,24)(16,21)(17,22)(18,27)(19,23)(20,26)(25,28);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,15)(14,20)(16,26)(17,19)(18,25)(21,22)(23,28)(24,27);(1,3,4,2)(6,9)(10,11)(13,16,19,18)(14,21,23,27)(15,25,17,26)(20,24,28,22);(5,7,8,12)(6,10,9,11)(13,17,19,15)(14,22,23,24)(16,26,18,25)(20,27,28,21);(13,16,19,18)(14,21,23,27)(15,26,17,25)(20,22,28,24)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371038);