Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2139 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)(5,6,10,11)(7,9,8,12)(15,19)(18,20)(21,22)(23,29)(24,26)(25,32)(27,31)(28,33)(30,35)(34,36);(5,7,10,8)(6,9,11,12)(14,17)(18,20)(22,24)(25,33)(27,29)(30,34);(6,11)(7,8)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,29)(25,34)(26,31)(28,36)(30,33)(32,35);(1,2,3,4)(5,8,10,7)(6,9,11,12)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,36,34,35);(5,9)(6,7)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5189070);