Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
2211 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)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17)(21,22)(23,24)(25,33)(26,27)(28,30)(29,36)(31,35)(32,34);(5,7,10,9)(6,11,8,12)(13,14,16,18)(15,17,20,19)(22,27)(23,30)(29,33)(31,34);(5,6,10,8)(7,12,9,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,28)(24,27)(25,34)(26,30)(29,32)(31,36)(33,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,23,27,30)(25,35,36,32)(29,34,33,31);(5,9,10,7)(6,11,8,12)(13,15)(14,17)(16,20)(18,19)(21,25)(22,29)(23,31)(24,32)(26,36)(27,33)(28,35)(30,34)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3269727);