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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(15,16)(19,20)(21,22,25,26)(23,27,29,33)(24,28,32,31)(30,34,35,36);(5,7,10,9)(6,11,8,12)(14,18)(19,20)(22,26)(23,29)(28,31)(30,35);(5,6,10,8)(7,12,9,11)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15,17,16)(14,19,18,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(5,9,10,7)(6,11,8,12)(13,16,17,15)(14,20,18,19)(21,24)(22,28)(23,30)(25,32)(26,31)(27,34)(29,35)(33,36)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fh,e^2=fh,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, 4521146);