Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
2181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,8)(7,11,12,10)(13,14,18,20)(15,17,16,19)(21,22)(23,26)(24,29)(25,30)(27,31)(28,33)(32,35)(34,36);(2,4)(5,7,9,12)(6,10,8,11)(13,15,18,16)(14,19,20,17)(22,23)(24,31)(25,33)(35,36);(1,2,3,4)(13,15,18,16)(14,17,20,19)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,36,34,35);(5,8,9,6)(7,11,12,10)(13,16,18,15)(14,17,20,19)(21,24)(22,27)(23,29)(25,34)(26,31)(28,36)(30,35)(32,33);(5,7)(6,11)(8,10)(9,12)(13,17)(14,16)(15,20)(18,19)(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,c^2=ghi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2767124);