Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
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 | |
2300 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)(7,11)(8,10)(9,12)(14,17)(15,19);(5,6,8,10)(7,9,12,11)(13,14,16,17)(15,20,19,18)(22,25)(26,28);(7,12)(9,11)(13,15,16,19)(14,18,17,20)(21,22,24,25)(23,26,27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,19)(17,20);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 468318);