Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2028 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)(5,6,8,10)(7,11,12,9)(13,14,18,16)(15,20,17,19)(22,26)(24,27);(7,12)(9,11)(13,14,18,16)(15,20,17,19)(21,22)(23,24)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15,18,17)(14,19,16,20)(21,22,25,26)(23,27,28,24);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,20,17,19)(21,23)(22,27)(24,26)(25,28);(5,7,8,12)(6,9,10,11)(13,17,18,15)(14,20,16,19)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3951625);