Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,10)(8,12)(13,14,18,16)(15,17,19,20)(21,22)(23,24)(25,33)(26,27)(28,29)(30,35)(31,34)(32,36);(5,6,9,10)(7,8,11,12)(14,16)(15,19)(22,27)(23,29)(30,33)(31,36);(5,7)(6,8)(9,11)(10,12)(13,15,18,19)(14,17,16,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,34)(32,33)(35,36);(1,2)(3,4)(5,8,9,12)(6,7,10,11)(13,16,18,14)(15,17,19,20)(21,24,26,28)(22,29,27,23)(25,34,35,32)(30,31,33,36);(5,7)(6,8)(9,11)(10,12)(13,17)(14,19)(15,16)(18,20)(21,25,26,35)(22,30,27,33)(23,31,29,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644419);