Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1842 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)(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22)(23,25)(24,32)(26,27)(28,31)(29,33)(30,36)(34,35);(5,6)(7,11)(8,10)(9,12)(14,17)(18,20)(22,27)(24,33)(25,28)(30,34);(5,6,8,10)(7,9,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,34)(25,27)(26,31)(29,35)(30,33)(32,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,24)(22,29)(23,30)(25,35)(26,33)(27,32)(28,36)(31,34);(5,7)(6,9)(8,12)(10,11)(13,15,16,19)(14,18,17,20)(21,25,26,28)(22,23,27,31)(24,35,33,36)(29,30,32,34)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408313);