Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| 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 | #15630 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2946 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)(15,16)(17,18)(19,20)(21,22)(23,25)(24,27)(26,28);(5,6,9,7)(8,11,12,10)(14,18)(16,19)(21,23,26,24)(22,27,28,25);(13,14,17,18)(15,19,20,16)(21,22)(23,25)(24,27)(26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,24,26,23)(22,27,28,25);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,15,18,20)(21,25,26,27)(22,23,28,24)Polycyclic presentation:
2^9|b^2=fg,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 560379);