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 | |
1833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(10,12)(13,14,17,19)(15,18,16,20)(21,22,26,30)(23,25,29,27)(24,32,35,28)(31,36,33,34);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,26,29)(22,27,30,25)(24,33,35,31)(28,34,32,36);(5,6)(7,10)(8,9)(11,12)(13,16,17,15)(14,18,19,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,36)(32,34)(33,35);(1,2,3,4)(13,14)(15,20)(16,18)(17,19)(21,24)(22,28)(23,31)(25,34)(26,35)(27,36)(29,33)(30,32);(5,7)(6,10)(8,11)(9,12)(21,25,26,27)(22,29,30,23)(24,34,35,36)(28,33,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=gi,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, 4571894);