Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2812 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)(8,12)(9,10)(13,14,16,17)(15,19,20,18)(22,26)(24,28);(6,10)(8,11)(13,14)(15,19)(16,17)(18,20)(21,22,25,26)(23,24,27,28);(6,10)(7,12)(14,17)(18,19)(22,26)(23,27);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28);(5,8)(6,7)(9,11)(10,12)(13,15,16,20)(14,18,17,19)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443006);