Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2974 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)(5,6,9,10)(7,8,11,12)(14,16)(15,19)(23,27)(26,28);(5,7)(6,8)(9,11)(10,12)(13,14,18,16)(15,20,19,17)(22,25)(26,28);(6,10)(7,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,16,18,14)(15,20,19,17);(5,8)(6,7)(9,12)(10,11)(13,17,18,20)(14,19,16,15)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4847889);