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 | |
2382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,22)(15,20,24,25)(16,23,26,27)(17,21,18,28);(2,4)(7,8)(10,12)(13,15)(14,20)(16,21)(17,27)(18,23)(19,24)(22,25)(26,28);(5,7,9,8)(6,10,11,12)(13,16,18,25,19,26,17,20)(14,15,21,23,22,24,28,27);(5,8,9,7)(6,10,11,12);(13,17,19,18)(14,21,22,28)(15,23,24,27)(16,20,26,25)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370132);