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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1975 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)(23,31)(24,34)(25,33)(26,27)(28,32)(29,35)(30,36);(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,18)(17,20)(22,27)(23,32)(24,35)(25,36);(5,6,8,9)(7,10,12,11)(13,14,16,18)(15,17,20,19)(21,23)(22,28)(24,30)(25,34)(26,32)(27,31)(29,36)(33,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,29)(23,33)(25,28)(26,35)(27,34)(30,32)(31,36);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,20)(18,19)(21,25)(22,30)(23,34)(24,28)(26,36)(27,33)(29,32)(31,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=gh,d^2=fi,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, 4407767);