Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| 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 | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2402 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)(7,10,11,12)(13,14)(15,19)(16,18)(17,20)(22,27)(24,29)(28,35)(32,33);(7,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,27)(23,29,31,24)(25,28,34,35)(30,33,36,32);(6,9)(10,12)(13,14,16,18)(15,17,20,19)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,35,33);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,33)(26,29)(27,31)(28,36)(30,35)(32,34);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,33)(31,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5518598);