Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,20,16)(23,27)(25,28);(2,4)(5,6,9,7)(8,11,12,10)(14,18)(16,19);(1,2,3,4)(6,7)(10,11)(13,15,17,20)(14,19,18,16)(21,22,24,26)(23,25,27,28);(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,23)(22,25)(24,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 612480);