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 | |
2077 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)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,19,20,17)(21,22)(23,30)(24,28)(25,35)(26,36)(27,29)(31,32)(33,34);(6,9)(10,11)(15,20)(17,19)(21,23,27,24)(22,28,29,30)(25,31,36,33)(26,32,35,34);(5,6)(7,11)(8,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,24,27,23)(22,28,29,30)(25,33,36,31)(26,32,35,34);(1,2,3,4)(13,15,16,20)(14,17,18,19)(21,25)(22,26)(23,31)(24,33)(27,36)(28,32)(29,35)(30,34);(5,7)(6,10)(8,12)(9,11)(21,26)(22,25)(23,32)(24,34)(27,35)(28,31)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=fhi,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, 4575306);