Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,17)(18,19)(23,24)(26,27);(6,9)(7,12)(13,15)(14,17)(16,18)(19,20)(21,22,25,28)(23,26,24,27);(6,9)(10,11)(14,19)(15,16)(21,23,25,24)(22,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,19,17)(21,22)(23,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,17,18,20)(14,15,19,16)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fg,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 494206);