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 | |
1847 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)(5,6,9,8)(7,11,12,10)(15,19)(18,20)(21,22,26,27)(23,25,30,28)(24,32,34,29)(31,33,36,35);(5,7,9,12)(6,10,8,11)(14,17)(18,20)(22,27)(23,30)(29,32)(31,36);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,30)(22,28,27,25)(24,31,34,36)(29,33,32,35);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,8,9,6)(7,11,12,10)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,36,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=fhi,e^2=f,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, 3375428);