Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,24,19,22,17,23)(15,25,26,16,21,27,28,20);(2,3)(6,9)(10,11)(13,15)(14,20)(16,22)(17,26)(18,28)(19,21)(23,25)(24,27);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16,19,20)(14,21,22,15)(17,25,18,27)(23,26,24,28);(1,3,4,2)(14,22)(15,21)(23,24)(26,28);(13,17,19,18)(14,23,22,24)(15,26,21,28)(16,27,20,25)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373335);