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 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,9)(7,10,11,12)(13,14,18,17)(15,16,20,19)(21,22)(23,31)(24,29)(25,35)(26,27)(28,32)(30,36)(33,34);(1,2,3,4)(6,9)(10,12)(13,15,18,20)(14,19,17,16)(22,27)(23,32)(29,33)(30,35);(1,2)(3,4)(7,11)(10,12)(13,16,18,19)(14,20,17,15)(21,23)(22,28)(24,30)(25,29)(26,32)(27,31)(33,36)(34,35);(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,29)(23,30)(25,28)(26,34)(27,33)(31,36)(32,35);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,16,20,19)(21,25)(22,30)(23,33)(24,31)(26,36)(27,35)(28,34)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=fgi,e^2=fg,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, 3340960);