Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1892 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)(5,6,9,11)(7,8,12,10)(14,17)(15,19)(22,26)(23,29)(24,32)(34,36);(5,7,9,12)(6,10,11,8)(13,14)(15,20)(16,17)(18,19)(24,32)(28,33)(31,35)(34,36);(6,11)(8,10)(13,14,16,17)(15,20,19,18)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2)(3,4)(5,8,9,10)(6,12,11,7)(13,14)(15,20)(16,17)(18,19)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fgh,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4069056);