Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1946 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)(5,6,9,12)(7,10,8,11)(13,14,18,16)(15,20,19,17)(21,22,26,23)(24,29,31,27)(25,28,34,30)(32,36,33,35);(5,7,9,8)(6,10,12,11)(15,19)(17,20)(22,23)(24,31)(28,30)(32,33);(5,7,9,8)(6,11,12,10)(13,15)(14,17)(16,20)(18,19)(21,23,26,22)(24,29,31,27)(25,28,34,30)(32,35,33,36);(1,2)(3,4)(5,8,9,7)(6,10,12,11)(13,16,18,14)(15,20,19,17)(21,24)(22,27)(23,29)(25,33)(26,31)(28,36)(30,35)(32,34);(5,6)(7,11)(8,10)(9,12)(13,17,18,20)(14,19,16,15)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gh,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3955058);