Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
1854 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)(5,6,9,7)(8,11,12,10)(14,16)(18,20);(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,18,19,20)(21,22,24,26)(23,27,28,25);(1,3,4,2)(6,7)(10,11)(15,19)(18,20)(21,23,24,28)(22,25,26,27);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 452791);