Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2158 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)(9,12)(10,11)(14,19)(15,17)(22,26)(23,29)(24,32)(34,36);(6,11)(7,9)(13,14,18,19)(15,20,17,16)(22,26)(24,32)(27,30)(31,35);(5,7,10,9)(6,12,11,8)(13,15,18,17)(14,20,19,16)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,9,10,7)(6,12,11,8)(13,17,18,15)(14,20,19,16)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|b^2=gh,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5515823);