Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,25,27)(23,24,28,26);(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,22,25,27)(23,24,28,26);(6,7)(10,11)(15,19)(17,20)(21,23,25,28)(22,26,27,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,20)(16,18)(17,19)(21,23)(22,24)(25,28)(26,27);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,17,18,20)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4068458);