Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2120 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)(15,24)(16,20)(17,21)(18,26)(19,22)(23,28)(25,27);(7,8)(10,12)(13,15,18,21,19,25,28,20)(14,17,23,24,22,16,26,27);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,23)(17,19)(18,27)(21,22)(24,28)(25,26);(5,8,9,7)(6,10,11,12)(13,17,19,16)(14,21,22,20)(15,23,25,26)(18,24,28,27);(6,11)(10,12)(14,22)(16,17)(23,26)(24,27)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390578);