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 | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(13,14,17,15)(16,19,20,18)(21,22)(23,27)(24,26)(25,28)(29,31)(30,35)(32,34)(33,36);(1,2,3,4)(7,12)(9,11)(22,24)(27,29)(28,32)(35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(5,7,8,12)(6,9,10,11)(13,16,17,20)(14,18,15,19)(21,24,26,22)(23,29,31,27)(25,28,34,32)(30,35,33,36);(5,6)(7,9)(8,10)(11,12)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,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, 3454666);