Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,21)(17,28)(18,22)(19,23)(24,26)(25,27);(1,2,3,4)(6,9)(10,11)(13,14)(15,24)(16,27)(17,22)(18,28)(19,23)(20,26)(21,25);(5,7,8,12)(6,10,9,11)(13,15,18,26,19,16,17,25)(14,20,22,24,23,21,28,27);(13,16,18,25,19,15,17,26)(14,21,22,27,23,20,28,24);(13,17,19,18)(14,22,23,28)(15,25,16,26)(20,24,21,27)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373691);