Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,11)(9,10)(13,14)(15,17)(16,18)(19,20)(21,22,27,30)(23,29,25,26)(24,28,33,35)(31,36,34,32);(5,6,9,10)(7,12,11,8)(13,14,16,18)(15,19,20,17)(21,23,27,25)(22,26,30,29)(24,31,33,34)(28,32,35,36);(6,10)(7,11)(13,14,16,18)(15,17,20,19)(21,24)(22,28)(23,31)(25,34)(26,32)(27,33)(29,36)(30,35);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,25,27,23)(22,29,30,26)(24,31,33,34)(28,32,35,36);(5,8)(6,11)(7,10)(9,12)(13,15)(14,17)(16,20)(18,19)(21,26,27,29)(22,25,30,23)(24,32,33,36)(28,34,35,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3045796);