Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,17,20)(15,18,19,16)(22,24)(27,28);(5,6,9,7)(8,11,12,10)(13,15,17,19)(14,18,20,16)(21,22,26,24)(23,27,25,28);(6,7)(10,11)(13,14)(15,18)(16,19)(17,20)(21,23,26,25)(22,27,24,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24,26,22)(23,27,25,28);(5,8)(6,10)(7,11)(9,12)(13,16,17,18)(14,19,20,15)(21,25,26,23)(22,27,24,28)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1002839);