Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,11)(7,9)(21,22,25,24)(23,27,28,26);(5,6)(7,8)(9,12)(10,11)(13,14,16,18)(15,17,19,20)(21,22,25,24)(23,27,28,26);(5,7,10,9)(6,12,11,8)(13,14)(15,17)(16,18)(19,20)(21,23,25,28)(22,26,24,27);(1,3,4,2)(5,8,10,12)(6,7,11,9);(5,9,10,7)(6,12,11,8)(13,15,16,19)(14,17,18,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=g,d^2=gi,e^2=h,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, 424316);