Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3358 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)(5,6,9,7)(8,11,12,10)(14,17)(18,20)(21,22,25,23)(24,27,28,26);(6,7)(10,11)(22,23)(26,27);(13,14)(15,18)(16,17)(19,20)(21,22,25,23)(24,26,28,27);(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,23,25,22)(24,27,28,26);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 429188);