Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2403 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,16)(17,19)(18,20)(22,23)(24,28);(6,10)(7,12)(13,15,18,17)(14,19,20,16)(22,23)(26,27);(6,10)(8,11)(14,20)(15,17)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,19)(14,15,20,17)(21,23,25,22)(24,27,28,26);(5,8,9,11)(6,7,10,12)(13,17,18,15)(14,19,20,16)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|b^2=fgh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507874);