Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,25)(23,29)(24,32)(34,36);(15,19)(17,20)(21,22,26,25)(23,30,29,27)(24,33,32,28)(31,34,35,36);(1,2)(3,4)(13,15,16,19)(14,17,18,20)(21,23,26,29)(22,27,25,30)(24,31,32,35)(28,34,33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24,26,32)(22,28,25,33)(23,31,29,35)(27,34,30,36);(5,7,8,11)(6,10,9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,22)(23,27,29,30)(24,33,32,28)(31,34,35,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=h,d^2=f,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4309977);