Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2117 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)(23,28)(24,29)(25,30)(26,27)(31,34)(32,33)(35,36);(5,6,8,10)(7,9,12,11)(14,16)(18,20)(22,27)(23,29)(30,36)(31,33);(5,7,8,12)(6,9,10,11)(13,14,17,16)(15,20,19,18)(21,23,26,29)(22,28,27,24)(25,33,35,31)(30,32,36,34);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,18,16,20)(21,24,26,28)(22,29,27,23)(25,34,35,32)(30,31,36,33);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,20,19,18)(21,25,26,35)(22,30,27,36)(23,31,29,33)(24,32,28,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=fg,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4064800);