Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2380 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)(8,9)(10,12)(13,14,18,19)(15,20,16,17)(23,27)(26,28);(6,9)(10,11)(14,19)(15,16)(21,22,25,24)(23,26,27,28);(13,15,18,16)(14,17,19,20)(21,23,25,27)(22,26,24,28);(5,6)(7,11)(8,9)(10,12)(13,16,18,15)(14,17,19,20);(5,7,8,12)(6,10,9,11)(13,17)(14,16)(15,19)(18,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426455);