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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2052 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)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,28)(25,26)(29,34)(30,33)(31,32)(35,36);(6,9)(10,12)(13,15,18,16)(14,19,17,20)(22,26)(27,33)(28,31)(34,35);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,19,17,20)(21,22)(23,27)(24,31)(25,26)(28,32)(29,35)(30,33)(34,36);(1,2)(3,4)(13,17,18,14)(15,19,16,20)(21,23,25,30)(22,27,26,33)(24,29,32,36)(28,34,31,35);(5,7)(6,10)(8,11)(9,12)(21,24,25,32)(22,28,26,31)(23,29,30,36)(27,34,33,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=fg,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, 4513784);