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×#2541 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3266 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)(7,11)(8,9)(10,12)(14,15)(18,19)(21,22)(23,27)(24,25)(26,28);(6,8)(10,11)(13,14,17,15)(16,19,20,18)(21,23,26,24)(22,27,28,25);(7,12)(10,11)(14,15)(18,19)(22,28)(23,24);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,25,28,27);(5,8,9,6)(7,11,12,10)(13,16,17,20)(14,18,15,19)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=fg,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.
| maximal subgroup of | #72263448, #72263453, #72263475 |
| maximal normal subgroup of | #72263448, #72263453, #72263475 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848484);