Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,24,31,28)(25,33,35,29)(30,36,34,32);(5,7,9,12)(6,10,11,8)(13,15,16,19)(14,17,18,20)(22,27)(24,28)(29,33)(32,36);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,28)(24,27)(25,34)(26,31)(29,32)(30,35)(33,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(21,24,26,28)(22,23,27,31)(25,32,35,36)(29,30,33,34);(5,8)(6,7)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,35)(22,29,27,33)(23,30,31,34)(24,32,28,36)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514362);