Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3515 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,12)(8,18)(9,14)(10,15)(11,16)(13,19)(17,20)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(1,2)(3,4)(5,7,11,9)(6,12,16,14)(8,17,19,10)(13,15,18,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,8,11,19)(6,13,16,18)(7,17,9,10)(12,15,14,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9,11,7)(6,14,16,12)(8,10,19,17)(13,20,18,15)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36);(5,10)(6,15)(7,8)(9,19)(11,17)(12,18)(13,14)(16,20);(5,6)(7,12)(8,13)(9,14)(10,20)(11,16)(15,17)(18,19)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565406);