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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2045 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)(6,9)(10,11)(13,14,16,18)(15,19,20,17)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(5,6)(7,10)(8,9)(11,12)(15,20)(17,19)(21,22,25,28)(23,26,31,29)(24,27,33,32)(30,34,36,35);(5,7,8,12)(6,10,9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,32)(25,28)(27,33)(29,31)(30,35)(34,36);(1,2,3,4)(13,15,16,20)(14,17,18,19)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,6)(7,11)(8,9)(10,12)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=hi,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, 4575296);