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, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2126 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)(5,6,9,7)(8,11,12,10)(13,14,17,15)(16,19,20,18)(22,25)(26,28);(6,7)(10,11)(13,14)(15,17)(16,18)(19,20)(23,27)(26,28);(1,3,4,2)(6,7)(10,11)(14,15)(18,19)(21,22)(23,26)(24,25)(27,28);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=f,e^2=fh,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, 452968);