Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2728 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)(7,10)(8,12)(9,11)(15,19)(17,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,19,20)(22,24)(27,29)(28,32)(34,35);(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769114);