Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1844 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)(7,8)(15,19)(17,20)(21,22,26,27)(23,28,32,34)(24,31,33,29)(25,30,35,36);(5,6,10,9)(7,12,8,11)(13,14,16,18)(15,17,19,20)(22,27)(24,33)(28,34)(30,36);(5,7,10,8)(6,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,30)(25,29)(26,32)(27,34)(31,35)(33,36);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,33)(22,29,27,31)(23,30,32,36)(25,34,35,28);(5,9,10,6)(7,11,8,12)(21,25)(22,30)(23,31)(24,34)(26,35)(27,36)(28,33)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2217722);