Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,16)(17,20)(18,19)(21,22,26,29)(23,28,31,25)(24,32,35,27)(30,36,33,34);(5,6,8,10)(7,9,12,11)(13,15)(14,18)(16,20)(17,19)(21,23,26,31)(22,25,29,28)(24,33,35,30)(27,34,32,36);(5,7,8,12)(6,9,10,11)(13,15)(14,16)(17,19)(18,20)(21,24,26,35)(22,27,29,32)(23,30,31,33)(25,36,28,34);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,23)(22,28)(24,30)(25,29)(26,31)(27,34)(32,36)(33,35);(5,7)(6,9)(8,12)(10,11)(13,16)(14,19)(15,20)(17,18)(21,25,26,28)(22,23,29,31)(24,34,35,36)(27,33,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4576988);