Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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×#3150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2254 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,20,24,19)(17,21,23,25)(22,26,28,27);(1,2,3,4)(6,9)(10,11)(14,18)(15,22)(17,23)(19,26)(20,27)(24,28);(5,7,8,12)(6,10,9,11)(13,15,23,28,16,24,17,22)(14,19,25,26,18,20,21,27);(13,15,17,22,16,24,23,28)(14,20,21,26,18,19,25,27);(14,18)(19,20)(21,25)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gh,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374006);