Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(14,18)(19,24)(20,25)(26,28);(1,2,3,4)(6,9)(10,11)(14,19)(15,21)(17,23)(18,24)(20,25)(22,27);(5,7,8,12)(6,10,9,11)(13,14,17,19,16,18,23,24)(15,20,21,26,22,25,27,28);(13,15,17,21,16,22,23,27)(14,20,19,26,18,25,24,28);(15,22)(20,25)(21,27)(26,28)Polycyclic presentation:
2^9|b^2=fgi,c^2=ghi,d^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373769);