Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1946 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)(7,8)(9,11)(10,12)(15,19)(18,20)(21,22,27,30)(23,25,24,28)(26,29,35,36)(31,33,32,34);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,34,36,33);(6,11)(8,10)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,28,30,25)(26,31,35,32)(29,33,36,34);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,27,28)(22,23,30,24)(26,34,35,33)(29,32,36,31);(5,8)(6,12)(7,11)(9,10)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,34)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375440);