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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2128 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)(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,19)(17,18)(21,22)(23,26)(24,32)(25,29)(27,31)(28,34)(30,36)(33,35);(6,9)(10,11)(14,17)(15,19)(21,23,25,31)(22,26,29,27)(24,33,34,30)(28,36,32,35);(6,9)(7,12)(13,15,18,19)(14,16,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,36)(32,35)(33,34);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,20)(14,19,17,15)(21,22)(23,26)(24,28)(25,29)(27,31)(30,35)(32,34)(33,36);(5,6)(7,11)(8,9)(10,12)(13,17,18,14)(15,16,19,20)(21,24)(22,28)(23,30)(25,34)(26,35)(27,36)(29,32)(31,33)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fgh,d^2=fgh,e^2=fgh,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, 4067639);