Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2490 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)(5,6,10,8)(7,9,12,11)(21,22,25,28)(23,26,24,27);(5,7,10,12)(6,11,8,9)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,26,24,27);(6,8)(7,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28);(5,8,10,6)(7,9,12,11)(21,23,25,24)(22,26,28,27);(5,9)(6,7)(8,12)(10,11)(13,15,16,19)(14,17,18,20)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423749);