Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1670 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)(5,6)(7,9)(8,10)(11,12)(21,22,27,25)(23,28,26,34)(24,31,33,29)(30,36,32,35);(6,7)(10,11)(14,17)(18,20)(21,23,27,26)(22,28,25,34)(24,30,33,32)(29,35,31,36);(8,12)(10,11)(13,14,16,17)(15,18,19,20)(21,24)(22,29)(23,30)(25,31)(26,32)(27,33)(28,35)(34,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(21,25,27,22)(23,28,26,34)(24,31,33,29)(30,35,32,36);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)(21,26,27,23)(22,28,25,34)(24,32,33,30)(29,35,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3261265);