Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2154 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)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,6,8,10)(7,9,12,11)(15,19)(18,20)(22,26)(27,29)(28,32)(34,35);(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,18,19,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(5,7,8,12)(6,9,10,11)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fgh,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, 2644397);