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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1967 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)(6,10)(7,12)(15,19)(18,20)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(5,6,9,10)(7,8,12,11)(14,17)(18,20)(22,26)(27,29)(28,32)(35,36);(6,10)(8,11)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,32)(25,26)(28,34)(29,31)(30,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35);(5,8,9,11)(6,12,10,7)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fh,d^2=fgh,e^2=f,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, 3252583);