Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(14,18)(15,20)(21,22)(23,29)(24,26)(25,32)(27,31)(28,34)(30,35)(33,36);(5,7,10,9)(6,8,11,12)(13,14,17,18)(15,19,20,16)(22,24)(27,29)(28,32)(35,36);(6,11)(7,9)(14,18)(16,19)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(5,9,10,7)(6,8,11,12)(13,16,17,19)(14,15,18,20)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4517337);