Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1749 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)(5,6,10,8)(7,11,9,12)(21,22,25,26)(23,27,31,29)(24,28,33,32)(30,34,36,35);(5,7,10,9)(6,11,8,12)(14,17)(18,20)(22,26)(27,29)(28,32)(34,35);(5,7,10,9)(6,12,8,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(5,9,10,7)(6,11,8,12)(13,15,16,19)(14,18,17,20)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=g,e^2=fgh,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, 2645550);