Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2549 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)(7,12)(8,10)(9,11)(13,14,19,16)(15,22,23,28)(17,21,18,27)(20,24,26,25);(7,8)(10,12)(13,15,17,24,19,23,18,25)(14,20,27,28,16,26,21,22);(1,2,3,4)(5,7,9,8)(6,10,11,12)(15,23)(20,26)(22,28)(24,25);(5,8,9,7)(6,10,11,12)(13,16,19,14)(15,20,23,26)(17,21,18,27)(22,24,28,25);(13,17,19,18)(14,21,16,27)(15,24,23,25)(20,22,26,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370306);