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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,7)(10,11)(13,14,17,15)(16,19,20,18)(21,22)(23,29)(24,31)(25,33)(26,27)(28,34)(30,32)(35,36);(2,4)(5,6,9,7)(8,11,12,10)(22,27)(24,29)(25,34)(30,36);(1,2)(3,4)(13,15,17,14)(16,19,20,18)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(5,7,9,6)(8,10,12,11)(13,16)(14,18)(15,19)(17,20)(21,24)(22,23)(25,35)(26,29)(27,31)(28,36)(30,33)(32,34);(5,8,9,12)(6,10,7,11)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fi,d^2=fg,e^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, 2796168);