Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C2×#2703 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2364 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)(6,7)(8,12)(15,19)(18,20)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(6,7)(10,11)(14,17)(18,20)(22,23)(25,33)(27,29)(31,36);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4408257);