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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(14,16)(18,20);(1,3,4,2)(6,10)(8,11)(15,19)(18,20)(22,25)(26,28);(5,7,9,12)(6,8,10,11)(13,14,17,16)(15,18,19,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23,24,27)(22,26,25,28);(5,8,9,11)(6,7,10,12)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=h,e^2=fg,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, 469107);