Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| 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×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2854 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)(7,11)(10,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(6,9)(10,12)(13,15,17,20)(14,18,19,16)(22,27)(23,28)(29,33)(30,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(1,2)(3,4)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215961);