Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
2336 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)(5,6,9,11)(7,8,12,10)(13,14,18,16)(15,20,19,17)(21,22,25,27)(23,28,26,24);(5,7,9,12)(6,10,11,8)(13,15,18,19)(14,17,16,20)(23,26)(24,28);(6,11)(7,12)(13,14,18,16)(15,20,19,17)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(5,8,9,10)(6,7,11,12)(13,16,18,14)(15,17,19,20);(5,7)(6,8)(9,12)(10,11)(13,17)(14,19)(15,16)(18,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029904);