Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2088 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)(5,6,9,10)(7,8,11,12)(13,14,17,18)(15,19,20,16)(21,22,25,24)(23,26,27,28);(6,10)(8,12)(13,15,17,20)(14,16,18,19)(23,27)(26,28);(6,10)(8,12)(13,14,17,18)(15,19,20,16)(21,23,25,27)(22,26,24,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,22,25,24)(23,26,27,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=h,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3781655);