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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(10,12)(14,15)(16,20)(21,22,26,27)(23,25,30,28)(24,29,33,35)(31,34,36,32);(1,2,3,4)(14,15)(18,19)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(13,14,17,15)(16,18,20,19)(21,23,26,30)(22,28,27,25)(24,31,33,36)(29,32,35,34);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,32)(30,36);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,28)(22,23,27,30)(24,32,33,34)(29,36,35,31)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=g,d^2=i,e^2=gi,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, 3636906);